Chapter 4.4, Problem 73E

(a)

To determine

To factor: given polynomial completely.

Answer to Problem 73E

The completely factored polynomial is $\left(7a+b\right)\left(c-d\right)\left(c+d\right)$ .

Explanation of Solution

Given information:

Given polynomial,

  $7a{c}^2+b{c}^2-7a{d}^2-b{d}^2$

Calculation:

Consider the polynomial $7a{c}^2+b{c}^2-7a{d}^2-b{d}^2$ :

Factor the polynomial as

  $\begin{array}{c}7a{c}^2+b{c}^2-7a{d}^2-b{d}^2=\left(7a{c}^2+b{c}^2\right)-\left(7a{d}^2+b{d}^2\right)\\ =c^2\left(7a+b\right)-d^2\left(7a+b\right)\\ =\left(7a+b\right)\left(c^2-d^2\right)\\ =\left(7a+b\right)\left(c-d\right)\left(c+d\right)\end{array}$

Therefore, the completely factored polynomial is $\left(7a+b\right)\left(c-d\right)\left(c+d\right)$ .

(b)

To determine

To factor: given polynomial completely.

Answer to Problem 73E

The completely factored polynomial is ${\left(x^n-1\right)}^2$ .

Explanation of Solution

Given information:

Given polynomial,

  $x^{2n}-2x^n+1$

Calculation:

Consider the polynomial $x^{2n}-2x^n+1$ :

Factor the polynomial as

  $\begin{array}{c}{x}^{2n}-2x^n+1={\left(x^n\right)}^2-2\cdot x^n+1^2\\ ={\left(x^n-1\right)}^2\end{array}$

Therefore, the completely factored polynomial is ${\left(x^n-1\right)}^2$ .

(c)

To determine

To factor: given polynomial completely.

Answer to Problem 73E

The completely factored polynomial is $\left(a^3-b^2\right){\left(ab+1\right)}^2$ .

Explanation of Solution

Given information:

Given polynomial,

  $a^5{b}^2-a^2{b}^4+2a^{4}b-2a{b}^3+a^3-b^2$

Calculation:

Consider the polynomial $a^5{b}^2-a^2{b}^4+2a^{4}b-2a{b}^3+a^3-b^2$ :

Factor the polynomial as

  $\begin{array}{c}{a}^5{b}^2-a^2{b}^4+2a^{4}b-2a{b}^3+a^3-b^2=\left(a^5{b}^2-a^2{b}^4\right)+\left(2a^4-2a{b}^3\right)+\left(a^2-b^2\right)\\ =a^2{b}^2\left(a^3-b^2\right)+2ab\left(a^3-b^2\right)+1\left(a^3-b^2\right)\\ =\left(a^3-b^2\right)\left(a^2{b}^2+2ab+1\right)\\ =\left(a^3-b^2\right){\left(ab+1\right)}^2\end{array}$

Therefore, the completely factored polynomial is $\left(a^3-b^2\right){\left(ab+1\right)}^2$ .