# The division of the rational expression, x + 3 4 x 2 − 9 ÷ x 2 + 7 x + 12 2 x 2 + 7 x − 15 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.4, Problem 31E
To determine

## To calculate: The division of the rational expression,  x+34x2−9÷x2+7x+122x2+7x−15

Expert Solution

The division of rational expression is (x+5)(2x+3)(x+4)

### Explanation of Solution

Given information:

The expression is given as:

x+34x29÷x2+7x+122x2+7x15

Formula used:

For the rational expression:

Fractions property for dividing rational expression:

AB÷CD=ABDC

Fractions property for multiplying rational expression:

ABCD=ACBD

Product formula: (A+B)(AB)=A2B2

Factoring trinomials: The factor of algebraic expression which contain three terms is of the from x2+bx+c ,

(x+r)(x+s)=x2+(r+s)x+rs

Choose the values of r and s which satisfied these equations r+s=b and rs=c

Calculation:

Consider the, algebraic expression

x+34x29÷x2+7x+122x2+7x15

Use the fraction property for dividing rational expression

AB÷CD=ABDC

x+34x29÷x2+7x+122x2+7x15=x+34x292x2+7x15x2+7x+12

Use the fraction property for multiplying rational expression

ABCD=ACBD

(x+3)(2x2+7x15)(4x29)(x2+7x+12)

Factor the second term in numeratorbyFactoring trinomials rule,

(x+3)(2x2+10x3x15)(4x29)(x2+7x+12)(x+3)(x+5)(2x3)(4x29)(x2+7x+12)

Factor the second term in denominator byFactoring trinomials rule and common 4 from the first term,

(x+3)(x+5)(2x3)(4x29)(x2+4x+3x+12)(x+3)(x+5)(x3)(4x29)(x+4)(x+3)

Apply the product formula in first term in denominator,

(x+3)(x+5)(2x3)(2x3)(2x+3)(x+4)(x+3)

Cancel the common factors from the numerator and denominator,

(x+5)(2x+3)(x+4)

Thus, the division of rational expression is (x+5)(2x+3)(x+4)

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