BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.3, Problem 73E
To determine

To calculate: The factors of the trinomial (3x+2)2+8(3x+2)+12 .

Expert Solution

Answer to Problem 73E

The factors of the trinomial (3x+2)2+8(3x+2)+12 are 3x+4 and 3x+8 .

Explanation of Solution

Given information:

The expression (3x+2)2+8(3x+2)+12 .

Formula used:

The special product formula of perfect square to multiply a polynomial with itself which is mathematically expressed as,

  (A+B)2=A2+2AB+B2

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 (3x+2)2+8(3x+2)+12 .

Recall the special product formula of perfect square to multiply a polynomial with itself which is mathematically expressed as,

  (A+B)2=A2+2AB+B2

Apply it,

  (3x+2)2+8(3x+2)+12=(3x)2+2(3x)(2)+22+8(3x)+82+12=9x2+12x+4+24x+16+12=9x2+36x+32

The simplified form of (3x+2)2+8(3x+2)+12 is 9x2+36x+32 .

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=9 , ps+qr=36 and rs=32 .

Here, p=3 , q=3 , r=8 and s=4 , so, 9x2+36x+32 will be factorized as,

  9x2+36x+32=9x2+(12+24)x+(8)(4)=9x2+12x+24x+32=3x(3x+4)+8(3x+4)=(3x+4)(3x+8)

Thus, the factors of trinomial (3x+2)2+8(3x+2)+12 are 3x+4 and 3x+8 .

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