Chapter 8.5, Problem 56PPS

(a)

To determine

The complete table.

Explanation of Solution

The table can be completed as:

  $\begin{array}{c}9x^2-24x+16=9x^2+mx-px-16\\ =9x^2-12x-12x-16\\ =\left(3x^2-12x\right)-\left(12x-16\right)\\ =3x(x-4)-4(3x-4)\\ =(3x-4)(3x-4)\end{array}$

  $\begin{array}{c}4x^2-20x+25=4x^2-10x-10x+25\\ =(4x^2-10)-(10x+25)\\ =2x(2x-5)-5(2x-5)\\ =(2x-5)(2x-5)\end{array}$

  $\begin{array}{c}16x^2+24x+9=16x^2+12x+12x+9\\ =(16x^2+12)+(12x+9)\\ =4x(4x+3)+3(4x+3)\\ =(4x+3)(4x+3)\end{array}$

  $\begin{array}{c}25x^2+20x+4=25x^2+10x+10x+4\\ =(25x^2+10)+(10x+4)\\ =5x(5x+2)+2(5x+2)\\ =(5x+2)(5x+2)\end{array}$

Polynomial	Factored Polynomial	First term	Last Term	Middle Term
$4x^2+12x+9$	(2x+3)(2x+3)	4x2=(2x)2	9=32	12x
$9x^2-24x+16$	(3x-4)(3x-4)	9x2=(3x)2	16=42	-24x
$4x^2-20x+25$	(2x-5)(2x-5)	4x2=(2x)2	25=52	-20x
$16x^2+24x+9$	(4x+3)(4x+3)	16x2=(4x)2	9=32	24x
$25x^2+20x+4$	(5x+2)(5x+2)	25x2=(5x)2	4=22	20x

(b)

To determine

The middle term of each polynomial.

Explanation of Solution

Given:

The given equation is:

  $4x^2+12x+9$

Calculation:

The solution for the given equation can be obtained as:

  $\begin{array}{l}\text{For }4x^2+12x+9,\text{ the middle term is 12}x\\ \text{For }9x^2-24x+16,\text{ the middle term is -24}x\\ \text{For }4x^2-20x+25,\text{ the middle term is -20}x\\ \text{For }16x^2+24x+9,\text{ the middle term is 24}x\\ \text{For }25x^2+20x+4,\text{ the middle term is 20}x\end{array}$

(c)

To determine

The pattern for a perfect square.

Explanation of Solution

Given:

The given equation is:

  $4x^2+12x+9$

Calculation:

The solution for the given equation can be obtained as:

  $\begin{array}{l}\left(a+b\right)\left(a+b\right)=a^2+2ab+b^{2}and\\ \left(a-b\right)\left(a-b\right)=a^2-2ab+b\end{array}$

(d)

To determine

The necessary conditions.

Explanation of Solution

The first and last terms must be perfect squares and the middle term must be 2 times the square roots of first and last terms.