# The factors of the trinomial 2 ( a + b ) 2 + 5 ( a + b ) − 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 74E
To determine

## To calculate: The factors of the trinomial 2(a+b)2+5(a+b)−3 .

Expert Solution

The factors of the trinomial 2(a+b)2+5(a+b)3 are (2a+2b1) and (a+b+3) .

### Explanation of Solution

Given information:

The expression 2(a+b)2+5(a+b)3 .

Formula used:

To find the factor of the trinomial of the form ax2+bx+c , find the numbers p,q,r and s such that sum of product of two numbers is equal to coefficient of x (ps+qr=b) , product of p and q is equal to the coefficient of x2 pq=a and product of r and s numbers is equal to constant term (rs=c) , such that (px+r) and (qx+s) are the factors of ax2+bx+c .

ax2+bx+c=pqx2+(ps+qr)x+rs=pqx2+psx+qrx+rs=px(qx+s)+r(qx+s)=(px+r)(qx+s)

Calculation:

Consider the given expression 2(a+b)2+5(a+b)3 .

Recall that to find the factor of the trinomial of the form ax2+bx+c , find the numbers p,q,r and s such that sum of product of two numbers is equal to coefficient of x (ps+qr=b) , product of p and q is equal to the coefficient of x2 (pq=a) and product of r and s numbers is equal to constant term (rs=c) , such that (px+r) and (qx+s) are the factors of ax2+bx+c .

ax2+bx+c=pqx2+(ps+qr)x+rs=pqx2+psx+qrx+rs=px(qx+s)+r(qx+s)=(px+r)(qx+s)

So, find p,q,r and s such that pq=2 , ps+qr=5 and rs=3 .

Here, p=2 , q=1 , r=1 and s=3 , so, 9x2+36x+32 will be factorized as,

2(a+b)2+5(a+b)3=2(a+b)2+(61)(a+b)+(1)(3)=2(a+b)2+6(a+b)(a+b)3=(2a+2b)(a+b+3)1(a+b+3)=(2a+2b1)(a+b+3)

Thus, the factors of trinomial 2(a+b)2+5(a+b)3 are (2a+2b1) and (a+b+3) .

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