The division of the rational expression, 4 y 2 − 9 2 y 2 + 9 y − 18 ÷ 2 y 2 + y − 3 y 2 + 5 y − 6

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 34E
To determine

To calculate: The division of the rational expression,  4y2−92y2+9y−18÷2y2+y−3y2+5y−6

Expert Solution

The division of rational expression is 1.

Explanation of Solution

Given information:

The expression is given as:

4y292y2+9y18÷2y2+y3y2+5y6

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

4y292y2+9y18÷2y2+y3y2+5y6

Use the fraction property for dividing rational expression

AB÷CD=ABDC

4y292y2+9y18÷2y2+y3y2+5y6=4y292y2+9y18y2+5y62y2+y3

Use the fraction property for multiplying rational expression

ABCD=ACBD

(4y29)(y2+5y6)(2y2+9y18)(2y2+y3)

Factor the second term in numerator by Factoring trinomials rule and apply product rule on first term,

(2y3)(2y+3)(y2+6yy6)(2y2+9y18)(2y2+y3)(2y3)(2y+3)(y+6)(y1)(2y2+9y18)(2y2+y3)

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

(2y3)(2y+3)(y+6)(y1)(2y2+12y3y18)(2y22y+3y3)(2y3)(2y+3)(y+6)(y1)(y+6)(2y3)(2y+3)(y1)

Cancel the common factors from the numerator and denominator,

All cut off only 1 left.

Thus, the division of rational expression is 1.

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