To simplify x 2 − 2 x − 3 x 2 + 8 x + 16 . 3 x + 12 x − 1 by using appropriate operations.

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, Problem 55RE
To determine

To simplify x2−2x−3x2+8x+16.3x+12x−1 by using appropriate operations.

Expert Solution

Explanation of Solution

Given information:

x22x3x2+8x+16.3x+12x1

Formula used:

Factoring Trinomials:- To factor the form x2+bx+c

(x+r)(x+s)=x2(r+s)x+rs where r+s=b and rs=c .

Property of Fractions: AB.CD=ACBD

Calculations:

By factoring both numerator and denominator we get,

First Term x22x3x2+8x+16

Numerator = x2+2x3

where r=3 and s=1 .

r+s=31=2rs=(3)(1)=3(x+r)(x+s)=(x+3)(x1)  (i)

Denominator = x2+8x+16

where r=4 and s=4 .

r+s=4+4=8rs=(4)(4)=16(x+r)(x+s)=(x+4)(x+4)  (ii)

Second Term 3x+12x1

=3(x+4)x1 (iii)

Now, dividing equation (i) and (ii) and multiplying equation (iii) we get,

=(x+3)(x1)(x+4)(x+4).3(x+4)(x1)

=3(x+3)(x1)(x+4)(x+4)(x+4)(x1)

=3(x+3)(x+4) ( by cancelling (x1)(x+4) )

After simplifying we get 3(x3)(x+4) .

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