Chapter 4.4, Problem 22E

To determine

To describe: and correct the error in factoring the given polynomial.

Answer to Problem 22E

The correct factorized form is $x^9+8x^3=x^3\left(x^2+2\right)\left(x^4-2x^2+4\right)$ .

Explanation of Solution

Given information:

A polynomial is given as

  $x^9+8x^3$

The polynomial is factorized as

  $\begin{array}{c}{x}^9+8x^3={\left(x^3\right)}^3+{\left(2x\right)}^3\\ =\left(x^3+2x\right)\left\{{\left(x^3\right)}^2-2x\cdot x^3+{\left(2x\right)}^2\right\}\\ =\left(x^3+2x\right)\left(x^6-2x^4+4x^2\right)\end{array}$

Calculation:

Consider the given polynomial.

  $x^9+8x^3$

Take common $x^3$ from each term.

  $x^9+8x^3=x^3\left(x^6+8\right)$

Now, factorize $\left(x^6+8\right)$ using the polynomial identity $a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)$ as shown:

  $\begin{array}{c}\left(x^6+8\right)={\left(x^2\right)}^3+{\left(2\right)}^3\\ =\left(x^2+2\right)\left\{{\left(x^2\right)}^2-x^2\cdot 2+{\left(2\right)}^2\right\}\\ =\left(x^2+2\right)\left(x^4-2x^2+4\right)\end{array}$

Therefore, $x^9+8x^3=x^3\left(x^2+2\right)\left(x^4-2x^2+4\right)$ .