Exercise 13 Page &nbsp; 172

Factor out 5x

5x(x-2)-x+2

Factor out -1

5x(x-2)-1(x-2)

Factor out (x-2)

(5x-1)(x-2)

We will continue with the denominator in the first fraction. This has the form of a perfect square trinomial.

x^2+8x+16

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Factor

WritePow

Write as a power

x^2+8x+4^2

SplitIntoFactors

Split into factors

x^2+2(x)(4)+4^2

FacPosPerfectSquare

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

(x+4)^2

Let's continue with factoring the polynomials in the second fraction. First, we will do the numerator.

x^2+10x+24

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Factor

WriteSum

Write as a sum

x^2+6x+4x+24

Factor out x

x(x+6)+4x+24

Factor out 4

x(x+6)+4(x+6)

Factor out (x+6)

(x+4)(x+6)

Finally, we factor the denominator in the second fraction.

10x^2+13x-3

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Factor

WriteDiff

Write as a difference

10x^2+15x-2x-3

Factor out 5x

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

Factor out -1

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

Factor out (2x+3)

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

Now, we can substitute the factored expressions and simplify the product by finding Giant Ones.

(5x-1)(x-2)/(x+4)^2*(x+4)(x+6)/(5x-1)(2x+3)

MultFrac

Multiply fractions

(5x-1)(x-2)(x+4)(x+6)/(x+4)^2(5x-1)(2x+3)

Commutative Property of Multiplication

(x-2)(x+4)(x+6)(5x-1)/(x+4)^2(2x+3)(5x-1)

Write as a product of fractions

(x-2)(x+4)(x+6)/(x+4)^2(2x+3)*(5x-1)/(5x-1)

a/a=1

(x-2)(x+4)(x+6)/(x+4)^2(2x+3)*1

Identity Property of Multiplication

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

PowToProdTwoFac

a^2=a* a

(x-2)(x+4)(x+6)/(x+4)(x+4)(2x+3)

Commutative Property of Multiplication

(x-2)(x+6)(x+4)/(x+4)(2x+3)(x+4)

Write as a product of fractions

(x-2)(x+6)/(x+4)(2x+3)*(x+4)/(x+4)

a/a=1

(x-2)(x+6)/(x+4)(2x+3)*1

Identity Property of Multiplication

(x-2)(x+6)/(x+4)(2x+3)

We want to again simplify the given rational expression.

6x+3/2x-3 ÷ 3x^2-12x+15/2x^2-x-3

The first step is to factor the polynomials in the fractions. In the first fraction, there is no common factors that we can factor out. Therefore, let's move on to the numerator in the second fraction. We can begin by factoring out a 3.

3x^2-12x-15=3(x^2-4x-5) Now the polynomial can be factored by finding two integers that multiply to -5 and add to -4.

3(x^2-4x-5)

▼

Factor

WriteSum

Write as a sum

3(x^2-5x+x-5)

Factor out x

3(x(x-5)+x-5)

AddPar

Add parentheses

3(x(x-5)+(x-5))

Factor out (x-5)

3(x+1)(x-5)

We will continue by factoring the denominator in the second fraction.

2x^2-x-3

▼

Factor

WriteSum

Write as a sum

2x^2-3x+2x-3

Factor out x

x(2x-3)+2x-3

AddPar

Add parentheses

x(2x-3)+(2x-3)

Factor out (2x-3)

(x+1)(2x-3)

Now, we can substitute the factored expressions and simplify the product by finding Giant Ones.

6x+3/2x-3÷3(x+1)(x-5)/(x+1)(2x-3)

Commutative Property of Multiplication

6x+3/2x-3÷3(x-5)(x+1)/(2x-3)(x+1)

Write as a product of fractions

6x+3/2x-3÷3(x-5)/2x-3*(x+1)/(x+1)

a/a=1

6x+3/2x-3÷3(x-5)/(2x-3)*1

Identity Property of Multiplication

6x+3/2x-3÷3(x-5)/(2x-3)

DivFracByFracDiv

a/b÷c/d=a/b*d/c

6x+3/2x-3*2x-3/3(x-5)

MultFrac

Multiply fractions

(6x+3)(2x-3)/(2x-3)*3(x-5)

Commutative Property of Multiplication

(6x+3)(2x-3)/3(x-5)(2x-3)

Write as a product of fractions

6x+3/3(x-5)*2x-3/2x-3

a/a=1

6x+3/3(x-5)*1

Identity Property of Multiplication

6x+3/3(x-5)

Factor out 3

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

ReduceFrac

a/b=.a /3./.b /3.

2x+1/x-5

Since the fractions have the same denominator, the numerators can be added directly. Then we will simplify the expression by factoring out the greatest common factor.

5m+18/m+3+4m+9/m+3

AddFrac

Add fractions

5m+18+4m+9/m+3

CommutativePropAdd

Commutative Property of Addition

5m+4m+18+9/m+3

AddTerms

Add terms

9m+27/m+3