dynamath reviewer for basic algebra

8/22/2019 DynaMath Reviewer for Basic Algebra

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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


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


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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dynamath reviewer for basic algebra

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