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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 107 Page 154

Practice makes perfect

a When multiplying two fractions the numerator and denominator can be multiplied separately. Then we can factor out a Giant One.

(3x-1)(x+7)/4(2x-5)*10(2x-5)/(4x+1)(x+7)

Multiply fractions

(3x-1)(x+7)* 10(2x-5)/4(2x-5)(4x+1)(x+7)

CommutativePropMult

Commutative Property of Multiplication

10(3x-1)(x+7)(2x-5)/4(4x+1)(x+7)(2x-5)

Write as a product of fractions

10(3x-1)/4(4x+1)*(x+7)(2x-5)/(x+7)(2x-5)

a/a=1

10(3x-1)/4(4x+1)*1

Identity Property of Multiplication

10(3x-1)/4(4x+1)

a/b=.a /2./.b /2.

5(3x-1)/2(4x+1)

b Let's start by simplifying the numerator and denominator separately.

(m-3)(m+11)/(2m+5)(m-3)÷(4m-3)(m+11)/(4m-3)(2m+5)

CommutativePropMult

Commutative Property of Multiplication

(m-3)(m+11)/(m-3)(2m+5)÷(4m-3)(m+11)/(4m-3)(2m+5)

Write as a product of fractions

(m-3/m-3*m+11/2m+1)÷(4m-3/4m-3*m+11/2m+5)

a/a=1

(1*m+11/2m+1)÷(1*m+11/2m+5)

Identity Property of Multiplication

m+11/2m+5÷m+11/2m+5

To divide fractions, we take the reciprocal of the second fraction and multiply them together. m+11/2m+5÷m+11/2m+5 [0.5em] ⇕ [0.5em] m+11/2m+5*2m+5/m+11 Look at the numerator and denominator of the fractions — they're the same. Therefore, this expression can be simplified to 1.

c We have another expression to multiply. To simplify, let's factor all polynomials before we multiply.

2p^2+5p-12/2p^2-5p+3*p^2+8p-9/3p^2-10p-8 Let's factor the first one.

2p^2+5p-12

▼

Factor

Write as a difference

2p^2+8p-3p-12

Factor out 2p

2p(p+4)-3p-12

Factor out 3

2p(p+4)-3(p+4)

Factor out (p+4)

(2p-3)(p+4)

The numerator of the first fraction can be simplified to (2p-3)(p+4). (2p-3)(p+4)/2p^2-5p+3*p^2+8p-9/3p^2-10p-8 Let's factor the second.

2p^2-5p+3

▼

Factor

Write as a difference

2p^2-2p-3p+3

Factor out 2p

2p(p-1)-3p+3

Factor out 3

2p(p-1)-3(p-1)

Factor out (p-1)

(2p-3)(p-1)

The factored form of the denominator in the first fraction is (2p-3)(p-1). (2p-3)(p+4)/(2p-3)(p-1)*p^2+8p-9/3p^2-10p-8 Let's factor the third.

p^2+8p-9

▼

Factor

Write as a difference

p^2+9p-p-9

Factor out p

p(p+9)-p-9

FactorOutNegSign

Factor out a minus sign

p(p+9)-(p+9)

Factor out (p+9)

(p-1)(p+9)

In the second fraction, the numerator can be factored to (p-1)(p+9). (2p-3)(p-1)/(2p-3)(p-1)*(p-1)(p+9)/3p^2-10p-8 Let's factor the fourth.

3p^2-10p-8

▼

Factor

Write as a difference

3p^2+12p-2p-8

Factor out 3p

3p(p+4)-2p-8

Factor out 2

3p(p+4)-2(p+4)

Factor out (p+4)

(3p-2)(p+4)

Now we have the factored forms. (2p-3)(p-1)/(2p-3)(p-1)*(p-1)(p+9)/(3p-2)(p+4) Let's simplify this.

(2p-3)(p+4)/(2p-3)(p-1)*(p-1)(p+9)/(3p-2)(p+4)

Multiply fractions

(2p-3)(p+4)(p-1)(p+9)/(2p-3)(p-1)(3p-2)(p+4)

CommutativePropMult

Commutative Property of Multiplication

(2p-3)(p+4)(p-1)(p+9)/(2p-3)(p+4)(p-1)(3p-2)

Write as a product of fractions

(2p-3)(p+4)(p-1)/(2p-3)(p+4)(p-1)* p+9/3p-2

a/a=1

1* p+9/3p-2

Identity Property of Multiplication

p+9/3p-2

d We have one more fraction to divide.

4x-12/x^2+3x-10÷2x^2-13x+21/2x^2+3x-35 Looking at the equation, we can tell that the numerator in the first fraction can be factored as 4x-12 = 4(x-3). Now let's factor the rest of the polynomials. 4(x-3)/x^2+3x-10÷2x^2-13x+21/2x^2+3x-35Let's factor the first expression.

x^2+3x-10

▼

Factor

Write as a difference

x^2+5x-2x-10

Factor out x

x(x+5)-2x-10

Factor out 2

x(x+5)-2(x+5)

Factor out (x+5)

(x-2)(x+5)

The denominator in the first fraction is (x-2)(x+5). 4(x-3)/(x-2)(x+5)÷2x^2-13x+21/2x^2+3x-35 Let's factor the second expression.

2x^2-13x+21

▼

Factor

Write as a difference

2x^2-6x-7x+21

Factor out 2x

2x(x-3)-7x+21

Factor out 7

2x(x-3)-7(x-3)

Factor out (x-3)

(2x-7)(x-3)

The numerator in the second fraction can be factored to (2x-7)(x-3). 4(x-3)/(x-2)(x+5)÷(2x-7)(x-3)/2x^2+3x-35 Let's factor the third expression.

2x^2+3x-35

▼

Factor

Write as a difference

2x^2+10x-7x-35

Factor out 2x

2x(x+5)-7x-35

Factor out 7

2x(x+5)-7(x+5)

Factor out (x+5)

(2x-7)(x+5)

Now we have the factored form. 4(x-3)/(x-2)(x+5)÷(2x-7)(x-3)/(2x-7)(x+5) Let's simplify this.

4(x-3)/(x-2)(x+5)÷(2x-7)(x-3)/(2x-7)(x+5)

Write as a product of fractions

4(x-3)/(x-2)(x+5)÷(2x-7/2x-7*x-3/x+5)

a/a=1

4(x-3)/(x-2)(x+5)÷(1*x-3/x+5)

Identity Property of Multiplication

4(x-3)/(x-2)(x+5)÷x-3/x+5

DivFracByFracDiv

a/b÷c/d=a/b*d/c

4(x-3)/(x-2)(x+5)*x+5/x-3

Multiply fractions

4(x-3)(x+5)/(x-2)(x+5)(x-3)

CommutativePropMult

Commutative Property of Multiplication

4(x-3)(x+5)/(x-2)(x-3)(x+5)

Write as a product of fractions

4/x-2*(x-3)(x+5)/(x-3)(x+5)

a/a=1

4/x-2*1

Identity Property of Multiplication

4/x-2

Subchapter links

3.2.1 Investigating Rational Functions p.140-141

3.2.2 Simplifying Rational Expressions p.145-146

3.2.3 Multiplying and Dividing Rational Expressions p.148-149

3.2.4 Adding and Subtracting Rational Expressions p.153-154

3.2.5 Creating New Functions p.157-159