Exercise 43 Page 532

We want to simplify the given rational expression. To do so, let's recall the Negative Exponent Property. x^(- n)=1/x^n This property can be rewritten for cases where we have a negative exponent in the denominator. 1/x^(- n)=x^nLet's use the above to start simplifying our expression.

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

CommutativePropMult

Commutative Property of Multiplication

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

WriteProdFrac

Write as a product of fractions

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

Using the property, we can rewrite the rearranged expression. 2x+6/x^2+2x-3* 1/(x-1)^(- 1) ⇕ 2x+6/x^2+2x-3* (x-1)^1 Now, we will factor the expression as much as we can. Then, we will cancel out any common factors.

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

MoveRightFacToNum

a/c* b = a* b/c

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

ExponentOne

a^1=a

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

▼

Factor the numerator

FactorOut

Factor out 2

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

▼

Factor the denominator

WriteDiff

Write as a difference

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

FactorOut

Factor out x

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

FactorOut

Factor out - 1

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

FactorOut

Factor out (x+3)

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

▼

Simplify

CancelCommonFac

Cancel out common factors

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

SimpQuot

Simplify quotient

2

We simplified the given expression. Finally, we will identify the restrictions on the variables from the denominator of the simplified expression and from any other denominator used. For simplicity, we will use their factored forms.

Denominator	Restrictions on the Denominator	Restrictions on the Variable
(x-1)^(- 1)	x-1≠ 0	x≠ 1
(x+3)(x-1)	x+3≠ 0 and x-1≠ 0	x≠ -3 and x≠ 1

We found two unique restrictions on the variable. x≠ 1, x≠ -3