# The factor of the expression ( a 2 + 2 a ) 2 − 2 ( a 2 + 2 a ) − 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 1.3, Problem 124E
To determine

## To calculate: The factor of the expression (a2+2a)2−2(a2+2a)−3 .

Expert Solution

The factor of the expression (a2+2a)22(a2+2a)3 is (a+1)2(a1)(a+3) .

### Explanation of Solution

Given information:

The expression (a2+2a)22(a2+2a)3 .

Formula used:

To find the factor of the trinomial of the form x2+bx+c , find two numbers r and s such that sum of the numbers is equal to coefficient of x (r+s=b) and product of two numbers is equal to constant term (rs=c) , such that (x+r) and (x+s) are the factors of x2+bx+c .

x2+bx+c=x2+(r+s)x+rs=x2+rx+sx+rs=x(x+r)+s(x+r)=(x+r)(x+s)

Calculation:

Consider the given expression (a2+2a)22(a2+2a)3 .

Recall that to find the factor of the trinomial of the form x2+bx+c , find two numbers r and s such that sum of the numbers is equal to coefficient of x (r+s=b) and product of two numbers is equal to constant term (rs=c) , such that (x+r) and (x+s) are the factors of x2+bx+c .

x2+bx+c=x2+(r+s)x+rs=x2+rx+sx+rs=x(x+r)+s(x+r)=(x+r)(x+s)

So, find two numbers r and s such that r+s=2 and rs=3 .

Here, r=3 and s=1 , so, (a2+2a)22(a2+2a)3 will be factorized as,

(a2+2a)22(a2+2a)3=(a2+2a)2+(3+1)(a2+2a)3=(a2+2a)23(a2+2a)+1(a2+2a)3=(a2+2a)(a2+2a3)+1(a2+2a3)=(a2+2a+1)(a2+2a3)

Simplify the expression further as,

(a2+2a)22(a2+2a)3=(a2+2a+1)(a2+2a3)=(a2+a+a+1)(a2+3aa3)=(a(a+1)+1(a+1))(a(a+3)1(a+3))=(a+1)2(a1)(a+3)

Thus, the factor of expression (a2+2a)22(a2+2a)3 is (a+1)2(a1)(a+3) .

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