greatest common factor and factoring by grouping 6.1 1.list all possible factors for a given number....
TRANSCRIPT
Greatest Common Factor and Factoring by Grouping6.16.1
1. List all possible factors for a given number.2. Find the greatest common factor of a set of numbers or
monomials.3. Write a polynomial as a product of a monomial GCF and
a polynomial.4. Factor by grouping.
Factors: Expressions that are multiplied
Terms: Expressions that are added/subtracted
2(3) 2 factors
2 factors2x
5xy
x + y
3(x + y)
(x + y)(3x + 4)(2x + 1)
3 factors
0 factors, 2 terms
2 factors, the second factor has 2 terms
3 factors each with 2 terms
Factored form: A number or an expression written as a product of factors.
3(x + 5) Factored form
3x + 1 5
(3x + 2)(x + 5)
3x(x + 5) + 2(x + 5)
Not factored form
Not factored form
Factored form
36
24
3
2
Greatest Common Factor = 12
Greatest common factor (GCF): The largest natural number that divides all given numbers with no remainder.
Listing Method for Finding GCF
1. List all possible factors for each given number. 2. Find the largest factor common to all lists.
312
212
36
24
Prime Factorization Method for Finding GCF
1. Write the prime factorization of each given number in exponential form.
2. Write the prime factors common to all factorizations. 3. Each factor is raised to the smallest exponent.4. Multiply the factors in the factorization.
Note: If there are no common prime factors, then the GCF is 1.
22
13
32332236
32322224
123432GCF 12
Find the GCF of a4b4c5 and a3b7.
Common primes: a and b.
The smallest exponent for a is 3.
The smallest exponent for b is 4.
GCF = a3b4
Find the GCF: 45a3b and 30a2.
GCF = 3 • 5 • a2 = 15a2
Multiplication and factoring are opposite operations.
In multiplication, we know the factors and want to find the product.
2 · 5 = 10
In factoring, we know the product and want to find the factors.
10 = 2 · 5
You are used to using the distributive property.8(y – 3) = 8y – 24
Now we want to reverse the process.
8y – 24 = 8(y – 3)
Factor:
GCF:
xx 2418 2
xx 2418 2
6 (3 4)x x
ALWAYS check by distributing!
6x
x6xx
xx
624
618 2
Factor:
GCF:
2
2
2
322
6
6
6
126
xy
xy
xy
yxxy
232 612 xyyx
26xy
232 612 xyyx
126 2 xyxy
Factoring a Monomial GCF Out of a Polynomial
1. Find the GCF of the terms in the polynomial. 2. Rewrite the given polynomial as a product of the
GCF and the quotient of the polynomial and the GCF.
polynomial = GCFGiven polynomial
GCF
Factor:
GCF:
2 3 4 29 15 18 x yz x y x y
Make the first term in the parentheses positive.Factor out the negative.
3x2 y
Watch your signs!!!
2 3 4 29 15 18 x yz x y x y2 3 4 2
22 2 2
9 15 183
3 3 3
x yz x y x yx y
x y x y x y
2 23 3 5 6x y z x x y
Factor:
GCF:
5432 116 rpnm
Prime polynomial
Slide 6- 15Copyright © 2011 Pearson Education, Inc.
Factor out the GCF.
a)
b)
c)
d)
4 2 2 356 32 72 x y xy x y
3 24 14 8 18 xy x y y x
24 14 8 18 xy x y y xy
2 28 7 4 9 xy x y xy x
3 28 7 4 9 xy x y y x
6.1
Slide 6- 16Copyright © 2011 Pearson Education, Inc.
Factor by factoring out the GCF.
a)
b)
c)
d)
4 2 2 356 32 72 x y xy x y
3 24 14 8 18 xy x y y x
24 14 8 18 xy x y y xy
2 28 7 4 9 xy x y xy x
3 28 7 4 9 xy x y y x
6.1
Factor:
GCF:
5 8 5 a b b
5 8 5 a b b
5 8 b a
(b + 5) Not in factored form!!
5 b
558
55
5bb
bba
b
Factor: 14214 xyxx
yxx 214
Slide 6- 19Copyright © 2011 Pearson Education, Inc.
Factor out the GCF.
a) Already factored
b)
c)
d)
6.1
1929 mmmm
329 mm
129 mmm
39 2 mm
Slide 6- 20Copyright © 2011 Pearson Education, Inc.
Factor by factoring out the GCF.
a) Already factored
b)
c)
d)
6.1
1929 mmmm
329 mm
129 mmm
39 2 mm
12 + 3b + 8a + 2abFactor:
Factor by GroupingFactor by Grouping
4 terms suggest
Rewrite with a binomial GCF.
12 + 3b + 8a + 2abFactor:
1. Group only the first two terms.
12 + 3b + 8a + 2ab2. Factor this first group.
4b2a
3. Write this same binomial factor for the second group.
44b2a b4. Find the common factor for the last two terms.
434b2a b5. Determine the correct sign. Check your signs!
434b2a b6. Factor out the common binomial.
324 ab
bbyy 44y 34 Factor:
1. Group only the first two terms.
bbyy 44y 34 2. Factor this first group.
4y3 y3. Write this same binomial factor for the second group.
44y3 yy4. Find the common factor for the last two terms.
44y3 yby5. Determine the correct sign. Check your signs!
44y3 yby6. Factor out the common binomial.
byy 34
bbyy 44y 34
Factor:
4y3 y 4yb
byy 34
2222 55 yayxax
Factor:
52 ax 5a2y
225 yxa
2222 33 mpnnpm Factor:
2222 33 mpnnpm
If there are no common factors, switch the middle two terms and try again. Keep the sign with each term.
2222 33 mnpnpm
2222 33 mnpnpm
22 nmp 22 nm 3
322 pnm
Slide 6- 27Copyright © 2011 Pearson Education, Inc.
Factor by grouping.
Prime PolynomialPrime Polynomial
6.1
axaybybx
Factor: 22322 15151010 bbabba
325 2 bbab
GCF:
b5
babbab 33225 222
bab 22 ba 23 b5
( )
Slide 6- 29Copyright © 2011 Pearson Education, Inc.
Factor by grouping.
a)
b)
c)
d)
2 3 7 21 b bc b c
7 3 b c
3 7 b c b
3 7 c b
3 7 b c
6.1
Slide 6- 30Copyright © 2011 Pearson Education, Inc.
Factor by grouping.
a)
b)
c)
d)
2 3 7 21 b bc b c
7 3 b c
3 7 b c b
3 7 c b
3 7 b c
6.1