# The product of the rational expression, x 2 + 7 x + 12 x 2 + 3 x + 2 ⋅ x 2 + 5 x + 6 x 2 + 6 x + 9

### 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 29E
To determine

## To calculate: The product of the rational expression,  x2+7x+12x2+3x+2⋅x2+5x+6x2+6x+9

Expert Solution

The product of rational expression is (x+4)(x+2)

### Explanation of Solution

Given information:

The expression is given as:

x2+7x+12x2+3x+2x2+5x+6x2+6x+9

Formula used:

For the rational expression:

Fractions property:

ABCD=ACBD

Factoring trinomials: The factor of rational 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, rationalexpression

x2+7x+12x2+3x+2x2+5x+6x2+6x+9

Use the fraction property,

ABCD=ACBD

(x2+7x+12)(x2+5x+6)(x2+3x+2)(x2+6x+9)

Factor the terms in numerator by Factoring trinomials rule,

(x2+4x+3x+12)(x2+3x+2x+6)(x2+2x+1x+2)(x2+3x+3x+9)(x+4)(x+3)(x+3)(x+2)(x2+2x+1x+2)(x2+3x+3x+9)

Factor the terms in denominator by Factoring trinomials rule,

(x+4)(x+3)(x+3)(x+2)(x+2)(x+1)(x+3)(x+3)

Cancel common factors from the numerators and denominator,

(x+4)(x+2)

Thus, the product of rationalexpression is (x+4)(x+2)

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