dynamath reviewer for basic algebra
TRANSCRIPT
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DynaMath
Math Reviewer for Basic
Algebra
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PrefaceThis book, DynaMath was developed and modified
to give students support and enhancement in their basic
algebraic ability.
Problems and solutions are stated and explained
plainly to allow maximum understanding in the topic
The questions and lessons found in this book help
students discover and exercise math concepts and
properties.
~DynaMath Production Team
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Table of ContentsPreface....................................................2
Table of Contents....................................3
Operations with Signed Numbers........4-6
Fundamental Assumptions....................7-9
Absolute Value...................................10-13
Exponents............................................14-17
Fractions............................................18-20
Radicals..............................................21-24
Special Products & Factoring............25-32
Solutions to Solve It!.........................33-41
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LESSON ONE: Operations with Signed NumbersLets Review!
Signed Numbers are positive or negative numbers. These
are the numbers with the plus (+) or negative (-) signs.
ADDITION
To add number with the similar sign, add and prefix common
sign.
Ex. 7 -48
+29 +-1936 -67
To add number with the unlike sign, get the difference then
prefix the sign of the bigger number to the result.
Ex. 32 -89
+-29 + 31
+3 -58SUBTRACTION
To subtract signed numbers, change the sign of the
subtrahend and proceed as in addition.
Ex. -36 -36 67 67
- 27 +-27 --15 + 15-63 82
-84 -36 56 56
--12 + 27 - 27 +-27
- 9 29
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MULTIPLICATION
To multiply signed numbers with like sign multiply the
numbers and prefix a plus sign to the product.
Ex. +18 -34x +5 x -7
+90 +238
To multiply signed numbers with unlike signs, multiply the
numbers and prefix a minus sign to the product.
Ex. +12 -15x - 8 x+ 6
-96 -90
DIVISION
To divide signed numbers with like sign, divide the
numbers then prefix a plus sign to the quotient.
Ex. = +9
= +8
To divide signed numbers with unlike sign, divide
the numbers then prefix a minus sign to thequotient.
Ex.
= -6= -4
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Solve It!
1. 22
+-71
2.
3. 37
x -5
4.
5. -56
-- 10
6. -123
x -3
7. 1.5
+-1.2
8. -45x 25
9. 35
--12
10.
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LESSON TWO: Fundamental Assumptions1. Commutative Law of Addition
The sum of two numbers is the same in whatever
order they are added.
Ex. 4+2=2+4
2. Associative Law of Addition
The sum of three or more numbers is the same in
whatever way the numbers are grouped.
Ex. 2+4+6=(2+4)+6=2+(4+6)=(2+6)+4
3. Commutative Law of Multiplication
The product of two or more numbers is the same in
whatever order they are multiplied.
Ex. 5*6=6*5=30
4. Associative Law of Multiplication
The product of three or more numbers is the same in
whatever way the numbers are grouped.
Ex. wxyz=(wx)yz=(wy)xz=(wz)xy=(yz)wx
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5. Distributive Law of Multiplication w/ Respect to
Addition
The product of a number and the sum of other
numbers is the same as the sum if the product by
multiplying each of the other number by the firstnumber.
Ex. 12(5x+7yz)= 60x+84yz
6.Distributive Law of Division w/ Respect to Addition
The sum of two or more numbers or when thedifference between two numbers is divided by a
number, the divisor must operate upon each term in
the dividend.
Ex.
=
+
+
+
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Solve It!
Identify the fundamental assumptions represented.
a)Commutative Law of Addition
b)Associative Law of Addtionc)Commutative Law of Addition
d)Associative Law of Multiplication
e)Distributive Law of Multiplication w/ Respect to
Addition
f)Distributive Law of Division w/ Respect to Addition
A B C D E F
1. . 3a+2 = 2+3a
2. 4(5+2)=20+8
3. 56*2=2*56
4. 10+13+5=(10+13)+5
5. (8b)b=b(8b)
6. 2(a+b+c)=2a+2b+2c
7.10+12=12+10
8. 5*6=6*5
9. 2*5*4*6=(2*5)(4*6)10. 50+20=20+50
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LESSON THREE: Absolute ValueLets Review!
Absolute Value of a number is the number of units from 0
on the number line.
The Absolute Value of a positive number or of zero is the
number itself.
The Absolute Value of a negative number is found by
changing the sign number.
Example 1: Solve |2x - 1| + 3 = 6
Step 1: Isolate the
absolute value
|2x - 1| + 3 = 6
|2x - 1| = 3
Step 2: Is the number on
the other side of theequation negative?
No, its a positive
number, 3, so continueon to step 3
Step 3: Write two
equations without
absolute value bars
2x - 1 = 3 2x - 1 = -3
Step 4: Solve both
equations
2x - 1 = 3
2x = 4
x = 2
2x - 1 = -3
2x = -2
x = -1
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Example 2: Solve |3x - 6| - 9 = -3
Step 1: Isolate the
absolute value
|3x - 6| - 9 = -3
|3x - 6| = 6
Step 2: Is the
number on the
other side of the
equation
negative?
No, its a positive
number, 6, so continue
on to step 3
Step 3: Write two
equations without
absolute value bars
3x - 6 = 6 3x - 6 = -6
Step 4: Solve both
equations
3x - 6 = 6
3x = 12
x = 4
3x - 6 = -6
3x = 0
x = 0
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Example 3: Solve |5x + 4| + 10 = 2
Step 1: Isolate the
absolute value
|5x + 4| + 10 = 2
|5x + 4| = -8Step 2: Is the number on
the other side of the
equation negative?
Yes, its a negative
number, -8. There is no
solution to this problem.
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Solve It!1.||=2. | |=3. ||=4.||=5. | | 6.
||=
7. ||=8.||=9.| |=10. ||=
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LESSON FOUR: Exponents
Lets Review
The expression means a * a *a *a. This indicates the fourthpower of a. The number 4 is called the exponent of the pow
and a is the base.
If the expression has no exponent , it is understood that its
exponent is 1.
LAWS OF EXPONENTS
1. MultipicationWhen terms of like bases are multiplied, the bases are taken
one and the powers of each base are added.
Ex:
*=2. Division
When terms of like bases are divided, subtract th
powers and retain their common base in the numerator.
Ex.
=
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3. Power of a PowerSimply multiply the exponent of the numerator
and denominator
Ex.
=
4. Power of a Product
Simply multiply the exponents of the given terms
Ex.
(xy
5. Power of Quotient
Simply multiply the exponent of the numerator and
denominator.
=
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6. Zero Exponent
Any number raised to the zero exponent is always equal
to one.
=17. Negative Exponent
When a number or a term is raised to a negative
exponent, simply take the reciprocal and change the
negative exponent to a positive exponent.
= 8. Fractional Exponents
Terms involving fractional exponent indicate the root
extractions of the given terms.
=6
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Solve It!
1. ++ =2.(+)(+)=3. )(5) =
4.
=
5. -10 =6.)(5) =7. -=8. +=9. 10. =
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LESSON FIVE: FractionsLets Review!
Rational Expressions are algebraic expressions whose
numerator and denominator are polynomials.
Proper Fraction is a fraction whose numerator is less than
the denominator.
Improper Fraction is a fraction which the numerator is
greaterthan the denominator.
Mixed Fraction is the sum of a polynomial or monomial
and a fraction.
Complex Fraction consists of one or more fractions.
ADDITION & SUBTRACTION OF FRACTIONS
To add or subtract fractions with common denominators,
add or subtract the numerators then write the result over
the denominator.
Ex. - = To add or subtract fractions with unlike
denominators, reduce them to fractions with a
common denominator.
Ex. +
=
x5 +
x2 =
+
=
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MIXED FRACTIONS
To reduce a mixed fraction to its simplest form, write the
integral part as a fraction with one as the denominator
and proceed as in addition of fractions.
Ex. 6c + = + = =
MULTIPLICATION OF FRACTIONS
To multiply fractions, multiply the numerators and the
denominators then perform cancellation if possible.
Ex. x
=
DIVISION OF FRACTIONS
To divide a fraction by another fraction, find thereciprocal of the second fraction and multiply.
Ex.
=
x
=
or
COMPLEX FRACTIONS
Has one or more fractions in its numerator or denominator
or both.
Ex.
=
=
= x
=
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Solve It!
1. +
2. 3a +
3.
-
4.
3
5. 4x -
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LESSON SIX: RadicalsLets Review!
A radical is an expression of the form .
is called a radical sign,a is called the radicand, and n
is called the index.
FRACTIONAL EXPONENTS
Fractional exponents indicate the root extractions of
given terms.
Ex. = =
LAWS OF RADICALS
(
)
=
(
)
= a
= x = =
REMOVING FACTORS OF RADICALS
Removing factors of radicals involves numbers that can
be factored in order to have a perfect square.
Ex. = = = 2
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RESOLVING THE INDEX OF A RADICAL
Resolving the index of a radical reduces the index of the
given radical.
Ex.
=
= = = RATIONALIZING THE DENOMINATOR
In order to rationalize, multiply both numerator and
denominator by the value of the denominator. This will not
change the value.
Ex.
=
= =
ADDITION & SUBTRACTION OF RADICALS
Addition and subtraction of radicals can be done if the
radicals have the same index and radicand.
Ex. - = (17-10) = 719 + 9 = (19+9) = 28
MULTIPLICATION OF RADICALS
Radicals can also be multiplied by simply multiplying the
given radicands
Ex. x = x = = =
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DIVISION OF RADICALS
Division of radicals is done by rationalizing the
denominator and proceeding to multiplication.
Ex. = x = = or 2
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Solve It!
1.
2.
3.
4.
5.
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LESSON SEVEN: Special Products & FactoringLets Review!
Special products are algebraic expressions consisting of
monomials or any polynomial
TYPES OF SPECIAL PRODUCTS
PRODUCTS OF POLYNOMIALS BY MONOMIALS
In order to get the products of polynomials by monomials,
multiply the monomials to all items in the polynomials.
Ex. PRODUCT OF THE SUM & DIFFERENCE OF TWO TERMS
The product of the sum and difference of two terms is
equal to the square of the first term minus the square of
the second term.
Ex. SQUARE OF THE SUM OF TWO TERMS
The square of the sum of two terms is equal to the
square of the first term plus twice the product of thetwo terms plus the square of the second term.
Ex.
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SQUARE OF THE DIFFERENCE OF TWO TERMS
The square of the difference of two terms is equal to the
square of the first term minus twice the product of the two
terms plus the square of the second term.
Ex.
PRODUCT OF BINOMIALS HAVING SIMILAR TERMS
The product of binomials having similar terms is equal tothe algebraic product of the first term plus the algebraic
sum of the cross product plus the algebraic product of the
second term.
Ex.
CUBE OF THE SUM OF TWO TERMS
The cube of the sum of two terms is equivalent to the
cube of the first term plus the cube of the last term, plus
three times the product of the square of the first term and
the second term, plus three times the product of the first
term and the square of the second term.
Ex.
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CUBE OF THE DIFFERENCE OF TWO TERMS
The cube of the difference of two terms is equal to the
cube of the first term minus the cube of the last term
minus three times the product of the square of the first
term and the second term plus three times the product of
the first term and the square of the second term.
Ex.
SQUARE OF TRINOMIALS
The square of trinomials is equivalent to the sum of
the square of the first term plus the square of the
second term plus the square of the last term plus
twice the product of the first and second term plus
twice the product of the first and last term plustwice the product of the second and last term.
Ex.
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TYPES OF FACTORING
COMMON MONOMIAL
Factoring a common monomial is simply finding the
common common factor of the given term.
Ex.
ac+ad=a(c+d)
DIFFERENCE OF TWO SQUARES
The difference of two squares is equal to the productof the sum and difference of the two terms.
Ex.
-=(x+y)(x-y)TRINOMIAL PERFECT SQUARE
The factor of a trinomial perfect square is written in the
form TRINOMIAL OF THE FORM
Trinomial of the form is finding two sets of factors that
when multiplied is equal to the given expression.
Ex.
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SUM & DIFFERENCE OF CUBES
The sum of cubes is equivalent to the sum of the
first and last terms times the square of the first terms
minus the product of the first and last terms plus
the square of the last term.
Ex. The difference of cubes is equivalent to the
difference of the first and last terms times the
square of the first term plus the product of the firstand last terms plus the square of the last term.
Ex. SUM AND DIFFERENCE OF ODD/EVEN POWERS
1. Sum of Odd Powers
The sum of two odd powers is always divisible by the sum
of the numbers.
Ex. 2. Difference of Odd PowersThe difference of two odd powers is always divisible
by the difference of the numbers.
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Ex. 3. Difference of Even Powers
The difference of even powers is factored as the
difference of squares.
Ex.
4. Sum of Even Powers
The sum of even powers is not factorable as such. Note,
however, that:
GROUPING OF TERMS
Grouping of terms id used to simplify the given
expressions. But we need to group first the terms withsimilar variables for us to find the common factor or similar
term.
Ex.
POLYNOMIAL PERFECT SQUARE
The factors of the polynomial perfect square is the square
of the sum of three terms.
Ex.
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EXPRESSION REDUCIBLE TO THE DIFFERENCE OF SQUARES
There are instances, however, when a given expression
can be reduced to the different squares. This is done by
grouping of terms.
Ex. [ ][ ]
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Solve It!
1. . 3. 4. 5. 6. 7. 8. 9. 10.
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Solutionsto
Solve It!
esson One
1. 22+-71
Get the difference the prefix
the sign of the bigger number
to the result.
2. Divide the numbers then prefix
a minus sign to the quotient.
3. 37
x -5
Multiply the numbers and
prefix a minus sign to the
product.
4.
Divide the numbers then prefix
a minus sign to the quotient.
5. -56
22-71
-49
= -59
37
x -5
-185
= -9
-56
+ 10
-46
-123x -3
+ 369
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-- 10
Change the sign of the
subtrahend and proceed as in
addition.
6. -123x -3
Multiply the numbers and
prefix a plus sign to the
product.
7. 1.5+-1.2
Get the difference the prefix
the sign of the bigger number
to the result.
8. -45
x 25
Multiply the numbers and
prefix a minus sign to the
product.
9. 35
--12
Change the sign of the
subtrahend and proceed as in
addition.
10.
Divide the numbers then prefix
a plus sign to the quotient.
Lesson Two
Identify the fundamental assumptions represented.
a)Commutative Law of Addition
b)Associative Law of Addtion
c)Commutative Law of Addition
d)Associative Law of Multiplication
e)Distributive Law of Multiplication w/ Respect to Addition
f)Distributive Law of Division w/ Respect to Addition
1.5
- 1.2
+.3
= 74
35
+ 12
+47
-45
x 25-1125
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A B C D E F
1. . 3a+2 = 2+3a
2. 4(5+2)=20+8
3. 56*2=2*56
4. 10+13+5=(10+13)+5
5. (8b)b=b(8b)
6. 2(a+b+c)=2a+2b+2c
7.10+12=12+10
8. 5*6=6*5
9. 2*5*4*6=(2*5)(4*6)
10. = ++
Lesson Three
1.||= 10There is NO negative sign outside the absolute value symbol.
Remember: ABSOLUTE VALUE IS ALWAYS POSTIVE.
2. | |= -3Subtract the terms located INSIDE the absolute value symbol, find
the absolute value of the difference then place the minus sign.
3.
||= 36
Multiply the two terms inside the absolute value symbol then find
the absolute value.
4.||= -9There is a negative sign outside the absolute value symbol so find
the absolute value then place the minus symbol to the result.
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5. | | 10Subtract the terms located INSIDE the absolute value symbol, find
the absolute value of the difference.
6. ||= -1.0Subtract the terms located INSIDE the absolute value symbol, find
the absolute value of the difference then place the minus sign.
7. ||= -200Multiply the two terms inside the absolute value symbol then findthe absolute value then place the minus symbol to the product.
8.||= 12There is NO negative sign outside the absolute value symbol.9.| |= 9Subtract the terms located INSIDE the absolute value symbol, find
the absolute value of the difference.10. ||= -8.25Multiply the two terms inside the absolute value symbol then find
the absolute value then place the minus symbol to the product.
Lesson Four
1. ++= 100+125+16
=241
2.(+)(+)=(225+4)(4+16)
=(229)(20)
=4580
3. )(5)
=(1)(5)
=5
4.
=
=4
5. -10=100-10
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=90
6.)(5)=(125)(5)
=625
7. -=512-16
=496
8. +=36+81=117
9. = =1369
10. =25+4=29
Lesson Five
1. + =
=
=
2. 3a +
=
=
3.
-
=
= =
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4. 3=
=
=
5. 4x -
=
-
=
Lesson Six
1. =
x
=
2. =
3.
=
=
=
=
= =4.
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= = =
5. =
=
=
==
Lesson Seven
1. = = Find the common term thenfactor out the term .. =
= Use the FOIL method between
the two terms then combine the
like terms from the product.
3. = = Find the common term thenfactor out the term
.4. = = Uses the FOIL method betweenthe two terms then cancel the
opposite terms from the product.
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5. =
Find two factors that can be
multiplied to get . Then findtwo factors that can be
multiplied to get -2 that when
added can get .6. =
= = Use the FOIL method between
the two terms then combine
the like terms from the
product.
7.
=Find the common term of theequation.
8. =
= = = = First find the square of3x+4 then combine like
terms then multiply the
product to another 3x+4
then simplify the product
by combining like terms.
9. = Find two factors that can be
multiplied to get . Thenfind two factors that can
be multiplied to get 12
that when added can get.10. = The difference of cubes is
equivalent to the
difference of the first and
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last terms times the square
of the first term plus the
product of the first and last
terms plus the square of
the last term.
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Published 2013