Chapter 4, Problem 22CR

To determine

To prove: That $x+2$ is a factor of $f\left(x\right)$ . Also, determine the factor of $f\left(x\right)$ completely.

Answer to Problem 22CR

The required factor of the polynomial is $\left(x+2\right)\left(x-3\right)\left(x^2+3x+9\right)$ .

Explanation of Solution

Given information:

The polynomial $f\left(x\right)=x^4+2x^3-27x-54$

Calculation:

The given polynomial is $f\left(x\right)=x^4+2x^3-27x-54$ .

To check whether $x+2$ is a factor of the above polynomial, divide $f\left(x\right)$ by $x+2$ .

If we get the remainder as 0 by dividing $f\left(x\right)$ by $x+2$ ; then this means that $x+2$ is a factor of $f\left(x\right)$ .

  $x+2\begin{array}{c}\hfill x^3-27\\ \hfill \overline{)\begin{array}{l}{x}^4+2x^3-27x-54\\ \underset{\_}{x^4+2x^3}\\ -27x-54\\ \underset{\_}{-27x-54}\\ 0\end{array}}\end{array}$

It is seen that the remainder is 0.

Thus, it is shown that $x+2$ is a factor of $f\left(x\right)$ .

Now, write the polynomial as the factors:

  $\begin{array}{c}{x}^4+2x^3-27x-54=\left(x+2\right)\left(x^3-27\right)\\ =\left(x+2\right)\left(x^3-3^3\right)\end{array}$

To factor the term $\left(x^3-3^3\right)$ , use the formula $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$

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

Hence, the required factor of the polynomial is $\left(x+2\right)\left(x-3\right)\left(x^2+3x+9\right)$ .