Exercise 71 Page 189

Practice makes perfect

To factor an expression, we want to find the greatest common factor that's shared by both terms. In the given expression, we can see that both terms contains x. Therefore, we can factor out x.

bx+ax=x(b+a)

The greatest common factor of the terms in the expression is x. Therefore, we can factor out x.

x+ax=x(1+a)

To be able to simplify the fraction we must factor both the numerator and denominator. The numerator, we can factor by pulling out an a. The denominator, we can factor since it is a perfect square trinomial.

ax+a/x^2 + 2 x + 1

FactorOut

Factor out a

a(x+1)/x^2 + 2 x + 1

SplitIntoFactors

Split into factors

a(x+1)/x^2+2(x)(1) + 1

WritePow

Write as a power

a(x+1)/x^2+2(x)(1) + 1^2

FacPosPerfectSquare

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

a(x+1)/(x + 1)^2

Now we can simplify the fraction by pulling out a Giant 1.

ax+a/(x+1)^2

FactorOut

Factor out a

a(x+1)/(x+1)^2

PowToProdTwoFac

a^2=a* a

a(x+1)/(x+1)(x+1)

WriteProdFrac

Write as a product of fractions

a/x+1*x+1/x+1

QuotOne

a/a=1

a/x+1*1

Multiply

a/x+1

Like in previous parts, we simply the fraction by factoring the numerator and denominator. The numerator contains a difference of squares which means it can be factored. In the denominator, we can factor out the a.

x^2-b^2/ax+ab

FacDiffSquares

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

(x+b)(x-b)/ax+ab

FactorOut

Factor out a

(x+b)(x-b)/a(x+b)

Now we can simplify the fraction by pulling out a Giant 1.

(x+b)(x-b)/a(x+b)

CommutativePropMult

Commutative Property of Multiplication

(x-b)(x+b)/a(x+b)