# The product of the rational expression, x 2 + 2 x y + y 2 x 2 − y 2 ⋅ 2 x 2 − x y − y 2 x 2 − x y − 2 y 2 ### 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 30E
To determine

## To calculate: The product of the rational expression,  x2+2xy+y2x2−y2⋅2x2−xy−y2x2−xy−2y2

Expert Solution

The product of rational expression is (2x+y)(x2y)

### Explanation of Solution

Given information:

The expression is given as:

x2+2xy+y2x2y22x2xyy2x2xy2y2

Formula used:

For the rationalexpression:

Fractions property: ABCD=ACBD

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

Factoring trinomials: The factor of rationalexpression 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 therationalexpression,

(x2+2xy+y2)(2x2xyy2)(x2y2)(x2xy2y2)

Use the fraction property,

ABCD=ACBD

Factor the terms in numerator by Factoring trinomials rule,

(x2+xy+xy+y2)(2x22xy+xyy2)(x2y2)(x2xy2y2)(x+y)(x+y)(2x+y)(xy)(x2y2)(x2xy2y2)

Factor the terms in denominator by Factoring trinomials rule and product property,

(x+y)(x+y)(2x+y)(xy)(x+y)(xy)(x2+xy2xy2y2)(x+y)(x+y)(2x+y)(xy)(x+y)(xy)(x2y)(x+y)

Cancel common factors from the numerators and denominator,

(2x+y)(x2y)

Thus, the product of rationalexpression is (2x+y)(x2y)

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