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Core Connections Algebra 2, 2013
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Core Connections Algebra 2, 2013 View details
2. Section 3.2
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Exercise 105 Page 153

Practice makes perfect
a Let's look at the expression we are given.

(x-4)^3(2x-1)/(2x-1)(x-4)^2First, we can rearrange the factors to create a Giant One (a form of the number one) and simplify. Let's try it!

(x-4)^3(2x-1)/(2x-1)(x-4)^2
CommutativePropMult

Commutative Property of Multiplication

(x-4)^3(2x-1)/(x-4)^2(2x-1)
WriteProdFrac

Write as a product of fractions

(x-4)^3/(x-4)^2*2x-1/2x-1
QuotOne

a/a=1

(x-4)^3/(x-4)^2 * 1
IdPropMult

Identity Property of Multiplication

(x-4)^3/(x-4)^2

We can simplify this further using the Quotient of Powers Property.

(x-4)^3/(x-4)^2
DivPow

a^m/a^n= a^(m-n)

(x-4)^(3-2)
SubTerm

Subtract term

(x-4)^1
ExponentOne

a^1=a

x-4
b We are again given a rational expression.

7m^2-22m+3/3m^2-7m-6 To simplify, let's factor the numerator and denominator.

7m^2-22m+3
â–¼
Factor
WriteDiff

Write as a difference

7m^2-m-21m+3
FactorOut

Factor out m

m(7m-1)-21m+3
FactorOut

Factor out 3

m(7m-1)-3(7m-1)
FactorOut

Factor out (7m-1)

(m-3)(7m-1)

The numerator can be factored to (m-3)(7m-1). Now we factor the denominator.

3m^2-7m-6
â–¼
Factor
WriteSum

Write as a sum

3m^2-9m+2m-6
FactorOut

Factor out 3m

3m(m-3)+2m-6
FactorOut

Factor out 2

3m(m-3)+2(m-3)
FactorOut

Factor out (m-3)

(3m+2)(m-3)

Using the factored forms, we can now simplify the expression.

7m^2-22m+3/3m^2-7m-6
SubstituteExpressions

Substitute expressions

(m-3)(7m-1)/(3m+2)(m-3)
CommutativePropMult

Commutative Property of Multiplication

(7m-1)(m-3)/(3m+2)(m-3)
WriteProdFrac

Write as a product of fractions

7m-1/3m+2* m-3/m-3
QuotOne

a/a=1

7m-1/3m+2* 1
IdPropMult

Identity Property of Multiplication

7m-1/3m+2
c Let's look at the given fraction.
(z+2)^9(4z-1)^7/(z+2)^(10)(4z-1)^5 To simplify this expression we can use that a power divided by another power with the same base can be simplified by subtracting the exponents.

(z+2)^9(4z-1)^7/(z+2)^(10)(4z-1)^5
WriteProdFrac

Write as a product of fractions

(z+2)^9/(z+2)^(10)*(4z-1)^7/(4z-1)^5
DivPow

a^m/a^n= a^(m-n)

(z+2)^(-1)(4z-1)^2
NegOneExponentToFrac

a^(- 1)=1/a

1/(z+2)*(4z-1)^2
MoveRightFacToNumOne

1/b* a = a/b

(4z-1)^2/z+2
d The numerator and denominator of this equation are not factored completely.

(x+2)(x^2-6x+9)/(x-3)(x^2-4) Let's factor the all of the expressions in the numerator and denominator before simplifying.

x^2-6x+9
FacNegPerfectSquare

a^2-2ab+b^2=(a-b)^2

(x-3)^2
PowToProdTwoFac

a^2=a* a

(x-3)(x-3)

Now let's factor the expression from the denominator.

x^2-4
WritePow

Write as a power

x^2-2^2
FacDiffSquares

a^2-b^2=(a+b)(a-b)

(x+2)(x-2)

The second factor in the denominator can be factored to (x+2)(x-2). We can use these factored forms to simplify the original expression.

(x+2)(x^2-6x+9)/(x-3)(x^2-4)
SubstituteExpressions

Substitute expressions

(x+2)(x-3)(x-3)/(x-3)(x+2)(x-2)
CommutativePropMult

Commutative Property of Multiplication

(x+2)(x-3)(x-3)/(x+2)(x-3)(x-2)
WriteProdFrac

Write as a product of fractions

(x+2)(x-3)/(x+2)(x-3)*x-3/x-2
QuotOne

a/a=1

1*x-3/x-2
IdPropMult

Identity Property of Multiplication

x-3/x-2
Subchapter links expand_more
arrow_right 3.2.1 Investigating Rational Functions p.140-141
63
64
65
66
67
68
Investigating Rational Functions 69 (Page 141)
arrow_right 3.2.2 Simplifying Rational Expressions p.145-146
Simplifying Rational Expressions 78 (Page 145)
Simplifying Rational Expressions 79 (Page 145)
80
Simplifying Rational Expressions 81 (Page 145)
Simplifying Rational Expressions 82 (Page 146)
Simplifying Rational Expressions 83 (Page 146)
84
arrow_right 3.2.3 Multiplying and Dividing Rational Expressions p.148-149
90
91
Multiplying and Dividing Rational Expressions 92 (Page 148)
Multiplying and Dividing Rational Expressions 93 (Page 149)
94
95
Multiplying and Dividing Rational Expressions 96 (Page 149)
arrow_right 3.2.4 Adding and Subtracting Rational Expressions p.153-154
Adding and Subtracting Rational Expressions 102 (Page 153)
103
Adding and Subtracting Rational Expressions 104 (Page 153)
Adding and Subtracting Rational Expressions 105 (Page 153)
Adding and Subtracting Rational Expressions 106 (Page 153)
Adding and Subtracting Rational Expressions 107 (Page 154)
108
Adding and Subtracting Rational Expressions 109 (Page 154)
arrow_right 3.2.5 Creating New Functions p.157-159
Creating New Functions 113 (Page 157)
114
Creating New Functions 115 (Page 158)
Creating New Functions 116 (Page 158)
Creating New Functions 117 (Page 158)
118
Creating New Functions 119 (Page 158)
Creating New Functions 120 (Page 159)
121
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Creating New Functions 123 (Page 159)
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Creating New Functions 126 (Page 159)