2 the least common multiple and clearing the denominators
TRANSCRIPT
![Page 1: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/1.jpg)
The Least Common Multiple (LCM)
![Page 2: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/2.jpg)
The Least Common Multiple (LCM)We say m is a multiple of x if m can be divided by x.
![Page 3: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/3.jpg)
For example, 12 is a multiple of 2
The Least Common Multiple (LCM)We say m is a multiple of x if m can be divided by x.
![Page 4: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/4.jpg)
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)We say m is a multiple of x if m can be divided by x.
![Page 5: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/5.jpg)
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
xy2 is a multiple of x,
We say m is a multiple of x if m can be divided by x.
![Page 6: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/6.jpg)
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
We say m is a multiple of x if m can be divided by x.
![Page 7: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/7.jpg)
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
We say m is a multiple of x if m can be divided by x.
![Page 8: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/8.jpg)
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
We say m is a multiple of x if m can be divided by x.
![Page 9: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/9.jpg)
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
![Page 10: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/10.jpg)
The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
![Page 11: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/11.jpg)
The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
![Page 12: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/12.jpg)
The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6,
xy is the LCM of x and y,
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
![Page 13: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/13.jpg)
The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6,
xy is the LCM of x and y,
and that 12xy is the LCM of 4x and 6y.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
![Page 14: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/14.jpg)
The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6,
xy is the LCM of x and y,
and that 12xy is the LCM of 4x and 6y.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
We give two methods for finding the LCM.
I. The searching method
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
![Page 15: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/15.jpg)
The least common multiple (LCM) of two or more numbers is
the smallest common multiple of all the given numbers.
For example, 12 is a multiple of 2. 12 is also a multiple of 3,
4, 6 or 12 itself.
For example, 12 is the LCM of 4 and 6,
xy is the LCM of x and y,
and that 12xy is the LCM of 4x and 6y.
The Least Common Multiple (LCM)
A common multiple of two or more quantities is a multiple of
all the given quantities.
For example, 24 is a common multiple of 4 and 6,
We give two methods for finding the LCM.
I. The searching method
II. The construction method
xy2 is a multiple of x, xy2 is also a multiple of y or xy.
xy2 is a common multiple of x and y.
We say m is a multiple of x if m can be divided by x.
![Page 16: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/16.jpg)
(LCM) The Searching MethodMethods of Finding LCM
The Minimum Coverage and The Construction Method
![Page 17: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/17.jpg)
(LCM) The Searching Method
Example B. Find the LCM of 18, 24, 16.
Methods of Finding LCM
The Minimum Coverage and The Construction Method
![Page 18: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/18.jpg)
(LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
Methods of Finding LCM
The Minimum Coverage and The Construction Method
![Page 19: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/19.jpg)
(LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24:
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The largest one on the list.
![Page 20: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/20.jpg)
(LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The largest one on the list.
![Page 21: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/21.jpg)
(LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The 1st one
that can also be
divided by 16&18.The largest one on the list.
![Page 22: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/22.jpg)
(LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The 1st one
that can also be
divided by 16&18.
![Page 23: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/23.jpg)
(LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The construction method is based on the principle of building
"the minimal (smallest, least) coverage which is necessary
to fulfill assorted requirements or conditions."
The 1st one
that can also be
divided by 16&18.
![Page 24: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/24.jpg)
(LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The construction method is based on the principle of building
"the minimal (smallest, least) coverage which is necessary
to fulfill assorted requirements or conditions."
The 1st one
that can also be
divided by 16&18.
Following are two concrete examples that demonstrate
the principle of constructing the “the minimum coverage”.
![Page 25: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/25.jpg)
(LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
so the LCM of {18, 24, 16} is 144.
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The construction method is based on the principle of building
"the minimal (smallest, least) coverage which is necessary
to fulfill assorted requirements or conditions."
Following are two concrete examples that demonstrate
the principle of constructing the “the minimum coverage”.
We will construct the LCD utilizing this principle later.
The 1st one
that can also be
divided by 16&18.
![Page 26: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/26.jpg)
The minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list.
Methods of Finding LCM
![Page 27: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/27.jpg)
The minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Methods of Finding LCM
![Page 28: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/28.jpg)
Methods of Finding LCMThe minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?
![Page 29: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/29.jpg)
Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Methods of Finding LCMThe minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?
![Page 30: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/30.jpg)
Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Science-3 yr
Methods of Finding LCMThe minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?
![Page 31: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/31.jpg)
Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Science-3 yr Math-3yrs
Methods of Finding LCMThe minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?
![Page 32: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/32.jpg)
Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Science-3 yr Math-3yrs English-4 yrs Arts-1yr
Methods of Finding LCMThe minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?
![Page 33: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/33.jpg)
Science Math. English Arts
A 2 yrs 3 yrs 3 yrs
B 3 yrs 3 yrs 2 yrs
C 2 yrs 2 yrs 4 yrs 1 yrs
Science-3 yr Math-3yrs English-4 yrs Arts-1yr
Methods of Finding LCMThe minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. A student wants to take enough courses so she
meets the requirement to apply to colleges A, B, and C.
The following is a table of requirements for each college,
how many years of each subject does she need?
form the least amount of required classes needed to be taken.
![Page 34: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/34.jpg)
LCM and LCDHere is another example of fulfilling "the minimal" requirements.
![Page 35: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/35.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
LCM and LCDHere is another example of fulfilling "the minimal" requirements.
![Page 36: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/36.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
LCM and LCD
What’s left after
Apu took some
items.
Here is another example of fulfilling "the minimal" requirements.
![Page 37: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/37.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
Here is another example of fulfilling "the minimal" requirements.
![Page 38: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/38.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
Here is another example of fulfilling "the minimal" requirements.
![Page 39: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/39.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
Here is another example of fulfilling "the minimal" requirements.
![Page 40: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/40.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
The least amount of items in the box consist of 2 apples,
Here is another example of fulfilling "the minimal" requirements.
![Page 41: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/41.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
The least amount of items in the box consist of 2 apples,
5 bananas and 4 carrots.
Here is another example of fulfilling "the minimal" requirements.
![Page 42: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/42.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
The least amount of items in the box consist of 2 apples,
5 bananas and 4 carrots. Apu took 1 apple and 1 banana,
Here is another example of fulfilling "the minimal" requirements.
![Page 43: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/43.jpg)
Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
The least amount of items in the box consist of 2 apples,
5 bananas and 4 carrots. Apu took 1 apple and 1 banana,
Bolo took 2 carrots, and Cato took 2 banana and 1 carrot.
Here is another example of fulfilling "the minimal" requirements.
![Page 44: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/44.jpg)
But when the LCM is large, the listing method is cumbersome.
LCM and LCD
![Page 45: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/45.jpg)
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
![Page 46: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/46.jpg)
Example D. Construct the LCM of {8, 15, 18}.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
![Page 47: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/47.jpg)
To construct the LCM:
a. Factor each number completely
Example D. Construct the LCM of {8, 15, 18}.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
![Page 48: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/48.jpg)
To construct the LCM:
a. Factor each number completely
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
![Page 49: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/49.jpg)
To construct the LCM:
a. Factor each number completely
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
![Page 50: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/50.jpg)
To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
![Page 51: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/51.jpg)
To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor:
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
![Page 52: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/52.jpg)
To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23,
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
The LCM must be divisible by 8 = 23
so it must have the factor 23
![Page 53: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/53.jpg)
To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
The LCM must be divisible by 18 = 2*32
so it must have the factor 32
(we have enough 2’s already)
![Page 54: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/54.jpg)
To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32, 5,
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
The LCM must has the factor 5
to be divisible by 15 = 3*5
(we have enough 3’s already)
![Page 55: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/55.jpg)
To construct the LCM:
a. Factor each number completely
b. For each prime factor, take the highest power appearing in
the factorizations. The LCM is their product.
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
From the factorization select the highest degree of each prime
factor: 23, 32, 5, so the LCM{8, 15, 18} = 23*32
*5 = 8*9*5 = 360.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
![Page 56: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/56.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
Methods of Finding LCM
![Page 57: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/57.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM =
Methods of Finding LCM
![Page 58: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/58.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3
Methods of Finding LCM
![Page 59: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/59.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3
Methods of Finding LCM
![Page 60: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/60.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4
Methods of Finding LCM
![Page 61: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/61.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
![Page 62: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/62.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 – x – 2
![Page 63: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/63.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 – x – 2
Factor each quantity.
![Page 64: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/64.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
![Page 65: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/65.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
![Page 66: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/66.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM =
![Page 67: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/67.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)
![Page 68: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/68.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)
![Page 69: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/69.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 70: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/70.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 71: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/71.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 72: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/72.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4)
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 73: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/73.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 74: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/74.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 75: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/75.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
Taking the highest power of each factor to get the
LCM =
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 76: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/76.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
Taking the highest power of each factor to get the
LCM = x2
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 77: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/77.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
Taking the highest power of each factor to get the
LCM = x2(x + 2)
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 78: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/78.jpg)
b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
d. Construct the LCM of x4 – 4x2, x2 – 4x +4
Factor each quantity.
x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2)
x2 – 4x + 4 = (x – 2)2
Taking the highest power of each factor to get the
LCM = x2(x + 2)(x – 2)2
c. Construct the LCM of x2 – 3x + 2, x2 + x – 2
Factor each quantity.
x2 – 3x + 2 = (x – 2)(x – 1)
x2 + x – 2 = (x + 2)(x – 1)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)
![Page 79: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/79.jpg)
Clearing Denominator with LCD
![Page 80: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/80.jpg)
The LCD is used for clearing denominators to simplify fractional calculations.
Clearing Denominator with LCD
![Page 81: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/81.jpg)
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.
![Page 82: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/82.jpg)
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 83: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/83.jpg)
Example E.
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 84: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/84.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 85: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/85.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
(x – 2)(x + 4) ( )
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 86: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/86.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
x– 2 ( ) (x – 2)(x + 4)
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
(x – 2)(x + 4)
x + 1
( )
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 87: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/87.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
x– 2 ( ) (x – 2)(x + 4)
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
(x – 2)(x + 4)
x + 1
( )
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 88: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/88.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
x– 2 ( ) (x – 2)(x + 4)
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
(x – 2)(x + 4)
x + 1
( )
= (x + 1)(x + 4)
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 89: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/89.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
x– 2 ( ) (x – 2)(x + 4)
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
(x – 2)(x + 4)
x + 1
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 90: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/90.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
x– 2 ( ) (x – 2)(x + 4)
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
x + 4 2x – 1
(x – 2)(x + 4)
c. Simplify (x – 2)(x + 4)
x + 1
( )
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 91: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/91.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
x– 2 ( ) (x – 2)(x + 4)
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
x + 4 2x – 1
(x – 2)(x + 4)
c. Simplify (x – 2)(x + 4)
x + 1
( )
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
x + 4 2x – 1
(x – 2)(x + 4) ( )
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 92: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/92.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
x– 2 ( ) (x – 2)(x + 4)
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
x + 4 2x – 1
(x – 2)(x + 4)
c. Simplify (x – 2)(x + 4)
x + 1
( )
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
= (2x – 1)(x – 2)x + 4 2x – 1
(x – 2)(x + 4) ( )
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 93: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/93.jpg)
Example E.
Their LCD is (x – 2)(x + 4).
x– 2 ( ) (x – 2)(x + 4)
a. Find the LCD of x + 1 x – 2 , x + 4
2x – 1
b. Simplifyx + 1 x – 2
x + 4 2x – 1
(x – 2)(x + 4)
c. Simplify (x – 2)(x + 4)
x + 1
( )
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
= (2x – 1)(x – 2)
= 2x2 – 5x + 2 x + 4 2x – 1
(x – 2)(x + 4) ( )
Clearing Denominator with LCDThe LCD is used for clearing denominators to simplify fractional calculations. Likewise the LCD is utilized in the same manner and purposes for simplifying rational expression calculations.Here is a typical sequence of steps of rational operations.
![Page 94: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/94.jpg)
x– 2 –
x + 4 ( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
Clearing Denominator with LCD
![Page 95: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/95.jpg)
x– 2 –
x + 4 )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
Clearing Denominator with LCD
![Page 96: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/96.jpg)
x– 2 –
x + 4 ( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
(x + 4)
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
Clearing Denominator with LCD
![Page 97: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/97.jpg)
x– 2 –
x + 4 ( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
Clearing Denominator with LCD
![Page 98: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/98.jpg)
= (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2 –
x + 4 ( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
Clearing Denominator with LCD
![Page 99: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/99.jpg)
= (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2 –
x + 4 ( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
Clearing Denominator with LCD
![Page 100: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/100.jpg)
= (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2 –
x + 4 ( )(x – 2)(x + 4)
Clearing Denominator with LCD
d. Simplify x + 1 2x – 1
(x + 4) (x – 2)
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
= x2 + 5x + 4 – 2x2 + 5x – 2
![Page 101: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/101.jpg)
= (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2 –
x + 4 ( )(x – 2)(x + 4)
Clearing Denominator with LCD
d. Simplify x + 1 2x – 1
(x + 4) (x – 2)
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
= x2 + 5x + 4 – 2x2 + 5x – 2
= – x2 + 10x + 2
![Page 102: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/102.jpg)
= (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2 –
x + 4 ( )(x – 2)(x + 4)
Clearing Denominator with LCD
d. Simplify x + 1 2x – 1
(x + 4) (x – 2)
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
= x2 + 5x + 4 – 2x2 + 5x – 2
= – x2 + 10x + 2
x– 2 –
x + 4 x + 1 2x – 1
e. Combine
![Page 103: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/103.jpg)
= (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2 –
x + 4 ( )(x – 2)(x + 4)
Clearing Denominator with LCD
d. Simplify x + 1 2x – 1
(x + 4) (x – 2)
x – 2 –
x + 4 ( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
= x2 + 5x + 4 – 2x2 + 5x – 2
= – x2 + 10x + 2
x– 2 –
x + 4 x + 1 2x – 1
This is the next topic.
e. Combine
![Page 104: 2 the least common multiple and clearing the denominators](https://reader031.vdocuments.us/reader031/viewer/2022021923/5a648e087f8b9a94568b4b6b/html5/thumbnails/104.jpg)
Ex. A. Find the LCM of the given expressions.
Methods of Finding LCM
xy3, xy1. x2y, xy32. xy3, xyz3.
4xy3, 6xy4. x2y, xy3, xy25. 9xy3, 12x3y, 15x2y36.
(x + 2), (x – 1) 7. (x + 2), (2x – 4) 8. (4x + 12), (10 – 5x) 9.
(4x + 6), (2x + 3) 10. (9x + 18), (2x + 4) 11. (–x + 2), (4x – 8) 12.
x2(x + 2), (x + 2)13. (x – 1)(x + 2), (x + 2)(x – 2)14.
(x – 3)(x – 3), (3 – x)(x + 2)17. x2(x – 2), (x – 2)x18.
19. x2(x + 2), (x + 2)(–x – 2) 20. (x – 1)(1 – x), (–1 – x)(x + 1)
(3x + 6)(x – 2), (x + 2)(x + 1)15. (x – 3)(2x + 1), (2x + 1)(x – 2)16.
x2 – 4x + 4, x2 – 421. x2 – x – 6, x2 – x – 222.
x2 + 2x – 3, 1 – x225. –x2 – 2x + 3, x2 – x26.
x2 – x – 6, x2 – 923. x2 + 5x – 6, x2 – x – 624.
2x2 + 5x – 3, x – 4x327. x2 – 5x + 4, x3 – 16x28.
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Multiplication and Division of Rational Expressions
33. x – 2 x2 – 9
( –x + 1
x2 – 2x – 3 )
34. x + 3 x2 – 4
( –2x + 1
x2 + x – 2 )
35. x – 1 x2 – x – 6
( –x + 1
x2 – 2x – 3 )
36.x + 2
x2 – 4x + 3( – 2x + 1
x2 + 2x – 3 )
29. 4 – xx – 3
( –x – 1
2x + 3 ) * LCD
30. 3 – x x + 2
( –2x + 3 x – 3
)
Ex B. Find the LCD of the terms then expand and simplify
the following expressions. (These are from the last section.)
31. 3 – 4xx + 1
( –1 – 2x x + 3
)
32. 5x – 7 x + 5
( –4 – 5xx – 3
)
* LCD
* LCD
* LCD
* LCD
* LCD
* LCD
* LCD