Exercise 52 Page 533

We want to multiply the given rational expressions. x/x^2-2x-3 * 2x-6/x^2-4x+3 We will start by factoring the numerators and denominators, if possible. Let's start with the first expression.

x/x^2-2x-3

▼

Factor the denominator

WriteDiff

Write as a difference

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

FactorOut

Factor out x

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

FactorOut

Factor out -3

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

FactorOut

Factor out (x+1)

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

Now, let's factor the second expression.

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

▼

Factor the numerator

FactorOut

Factor out 2

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

▼

Factor the denominator

WriteDiff

Write as a difference

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

FactorOut

Factor out x

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

FactorOut

Factor out - 3

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

FactorOut

Factor out (x-1)

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

Now that both expressions have been factored, we will cancel out any common factors.

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

CancelCommonFac

Cancel out common factors

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

SimpQuot

Simplify quotient

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

CommutativePropMult

Commutative Property of Multiplication

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

The obtained expression corresponds to the Option C.