The division of the rational expression, 2 x 2 + 3 x + 1 x 2 + 2 x − 15 ÷ x 2 + 6 x + 5 2 x 2 − 7 x + 3

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 33E
To determine

To calculate: The division of the rational expression,  2x2+3x+1x2+2x−15÷x2+6x+52x2−7x+3

Expert Solution

The division of rational expression is (2x+1)(2x1)(x+5)(x+5)

Explanation of Solution

Given information:

The expression is given as:

2x2+3x+1x2+2x15÷x2+6x+52x27x+3

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

2x2+3x+1x2+2x15÷x2+6x+52x27x+3

Use the fraction property for dividing rational expression

AB÷CD=ABDC

2x2+3x+1x2+2x15÷x2+6x+52x27x+3=2x2+3x+1x2+2x152x27x+3x2+6x+5

Use the fraction property for multiplying rational expression

ABCD=ACBD

(2x2+3x+1)(2x27x+3)(x2+2x15)(x2+6x+5)

Factor the first and second terms in numerator by Factoring trinomials rule,

(2x2+2x+x+1)(2x26x1x+3)(x2+2x15)(x2+6x+5)(2x+1)(x+1)(x3)(2x1)(x2+2x15)(x2+6x+5)

Factor the first and second terms in denominator by Factoring trinomials rule,

(2x+1)(x+1)(x3)(2x1)(x2+5x3x15)(x2+5x+1x+5)(2x+1)(x+1)(x3)(2x1)(x+5)(x3)(x+5)(x+1)

Cancel the common factors from the numerator and denominator,

(2x+1)(2x1)(x+5)(x+5)

Thus, the division of rational expression is (2x+1)(2x1)(x+5)(x+5)

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