Chapter 6.8, Problem 60STP

To determine

To calculate: The factors of the polynomial $a^4-2a^3-17a^2+18a+72$ .

Answer to Problem 60STP

The factors of the polynomial $a^4-2a^3-17a^2+18a+72$ are $\left(a-4\right),\left(a+3\right)\text{,}\left(a+2\right)\text{ and }\left(a-3\right)$ .

Explanation of Solution

Given information:

One of the factor of the polynomial is $a-3$ .

Formula used:

In long division of polynomials $\text{Dividend}=\text{Divisor}\times \text{Quotient}+\text{Remainder}$ .

Calculation:

Consider thepolynomial $a^4-2a^3-17a^2+18a+72$ . One of the factor of the polynomial is $a-3$ .

So, $a-3$ is a factor of the polynomial $a^4-2a^3-17a^2+18a+72$ .

Use long division to divide $a^4-2a^3-17a^2+18a+72$ by its factor $a-3$ .

  $a-3\begin{array}{c}\hfill a^3+a^2-14a-24\\ \hfill \overline{)\begin{array}{l}\begin{array}{ccccccccc}{a}^4& -& 2a^3& -& 17a^2& +& 18a& +& 72\end{array}\\ \begin{array}{ccccccc}\begin{array}{cc}{a}^4& \end{array}-3a^3& & & & & & \end{array}\\ \begin{array}{ccccccc}& & a^3& -& 17a^2& & \end{array}\\ \begin{array}{ccccccc}& & a^3& -3a^2& & & \end{array}\\ \begin{array}{ccccccc}& & -14a^2& +& 18a& & \end{array}\\ \begin{array}{ccccccc}& & -14a^2& +& 42a& & \end{array}\\ \begin{array}{ccccccc}& & & & -24a+72& & \end{array}\\ \begin{array}{ccccccc}& & & & -24a+72& & \end{array}\\ \begin{array}{ccccccc}& & & & & 0& \end{array}\end{array}}\end{array}$

The depressed polynomial is $a^3+a^2-14a-24$ .

For the above polynomial its one of the root is $a=-2$ .

Use long division to divide $a^3+a^2-14a-24$ by its factor $a+2$ .

  $a+2\begin{array}{c}\hfill a^2-a-12\\ \hfill \overline{)\begin{array}{l}\begin{array}{ccccccc}{a}^3& +& a^2& -& 14a& -& 24\end{array}\\ \begin{array}{ccccccc}\begin{array}{cc}{a}^3& \end{array}+2a^2& & & & & & \end{array}\\ \begin{array}{ccccccc}& & -a^2& -& 14a& & \end{array}\\ \begin{array}{ccccccc}& & -a^2& -2a& & & \end{array}\\ \begin{array}{ccccccc}& & -12a& -& 24& & \end{array}\\ \begin{array}{ccccccc}& & -12a& -& 24& & \end{array}\\ \begin{array}{ccccccc}& & & & 0& & \end{array}\end{array}}\end{array}$

Factor the above polynomial,

  $\begin{array}{c}{a}^2-a-12=a^2-4a+3a-12\\ =a\left(a-4\right)+3\left(a-4\right)\\ =\left(a-4\right)\left(a+3\right)\end{array}$

Therefore, factors are $\left(a-4\right),\left(a+3\right)\text{,}\left(a+2\right)\text{ and }\left(a-3\right)$ .

Thus, the factors of the polynomial $a^4-2a^3-17a^2+18a+72$ are $\left(a-4\right),\left(a+3\right)\text{,}\left(a+2\right)\text{ and }\left(a-3\right)$ .