Chapter 8.7, Problem 41PPS

To determine

To factor the given polynomial

Answer to Problem 41PPS

  $x(x+6)(x-6)(x+6)$

Explanation of Solution

Given:

Polynomial: $x^4+6x^3-36x^2-216x$

Concept used:

Finding common factor in each term and factoring using grouping method.

Calculation:

Consider the given polynomial,

  $\Rightarrow x^4+6x^3-36x^2-216x$

Group the like terms and find common factor.

  $\Rightarrow (x^4+6x^3)-(36x^2+216x)$

Find out common terms in the above expression.

  $\begin{array}{l}\Rightarrow (x^4+6x^3)-(36x^2+216x)\\ \Rightarrow x^2(x^2+6x)-36(x^2+6x)\end{array}$

Here, $(x^2+6x)$ is a common factor.

  $\Rightarrow (x^2-36)(x^2+6x)$

Further, $(x^2-36)$ can be factored as,

  $\begin{array}{l}$ \Rightarrow (x^2-36)=({(x)}^2-6^2)=(x-6)(x+6)$\because a^2-b^2=(a+b)(a-b)\end{array}$

Also, $(x^2+6x)$ has x as a common factor,

  $\Rightarrow x^2+6x=x(x+6)$

Therefore, the polynomial can be factorised as,

  $\Rightarrow x(x+6)(x-6)(x+6)$

Conclusion:

Therefore, the polynomial is factored as $x(x+6)(x-6)(x+6).$