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Pearson Algebra 2 Common Core, 2011

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Pearson Algebra 2 Common Core, 2011 View details

4. Ellipses

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Exercise 71 Page 644

Factor the numerator and denominator as much as possible. Cancel out common factors, if possible.

Simplified Expression: x-5/x^2-2x+4
Restrictions: x ≠ - 2

Practice makes perfect

We want to simplify the given rational expression. To do so, we will factor the numerator and denominator as much as possible. Then, we will cancel out any common factors.

x^2-3x-10/x^3+8

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Factor the numerator

Write as a sum

x^2-5x+2x-10/x^3+8

Factor out x

x(x-5)+2x-10/x^3+8

Factor out 2

x(x-5)+2(x-5)/x^3+8

Factor out (x-5)

(x-5)(x+2)/x^3+8

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Factor the denominator

Identity Property of Addition

(x-5)(x+2)/x^3+0+8

Rewrite 0 as 2x^2-2x^2

(x-5)(x+2)/x^3+2x^2-2x^2+8

Identity Property of Addition

(x-5)(x+2)/x^3+2x^2-2x^2+0+8

Rewrite 0 as 4x-4x

(x-5)(x+2)/x^3+2x^2-2x^2+4x-4x+8

CommutativePropAdd

Commutative Property of Addition

(x-5)(x+2)/x^3+2x^2-2x^2-4x+4x+8

Factor out x^2

(x-5)(x+2)/x^2(x+2)-2x^2-4x+4x+8

Factor out - 2x

(x-5)(x+2)/x^2(x+2)-2x(x+2)+4x+8

Factor out 4

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

Factor out (x+2)

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

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Simplify

CancelCommonFac

Cancel out common factors

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

Simplify quotient

x-5/x^2-2x+4

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. Let's try to factor the denominator using the Quadratic Formula with a= 1, b= - 2, and c= 4.

x=- b±sqrt(b^2-4ac)/2a

SubstituteValues

Substitute values

x=- ( - 2)±sqrt(( - 2)^2-4( 1)( 4))/2( 1)

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Evaluate right-hand side

- (- a)=a

x=2±sqrt((- 2)^2-4(1)(4))/2(1)

Calculate power and product

x=2±sqrt(4-16)/2

Subtract terms

x=2±sqrt(- 12)/2 *

The square root of a negative number is not defined in real numbers. This means that the denominator is already written in its simplest form.

Denominator	Restrictions on the Denominator	Restrictions on the Variable
(x+2)(x^2-2x+4)	x+2 ≠ 0 and x^2-2x+4 ≠ 0	x ≠ - 2
x^2-2x+4	x^2-2x+4 ≠ 0	-

We found one unique restriction on the variable. x ≠ - 2

Subchapter links

Lesson Check p.642

Practice and Problem-Solving Exercises p.642-644

Standardized Test Prep p.644

Mixed Review p.644