Chapter 12.4, Problem 51STP

To determine

To find: The factored form of the given expression.

Answer to Problem 51STP

Factored form of given expression is

  $f\left(x\right)=\left(x-4\right)\left(x+3\right)\left(x-6\right)$

Explanation of Solution

Given information:

  $f\left(x\right)=x^3-7x^2-6x+72$

  $4$ is one of the zeros of $f\left(x\right)$ .

Given

  $f\left(x\right)=x^3-7x^2-6x+72$

  $4$ is one of the zeros of $f\left(x\right)$ .

So, $f\left(x\right)$ can be written as

  $x^3-7x^2-6x+72=\left(x-4\right)\left(a{x}^2+bx+c\right)$

On simplifying and comparing

We get

  $\begin{array}{l}{x}^3-7x^2-6x+72=\left(x-4\right)\left(a{x}^2+bx+c\right)\\ \Rightarrow x^3-7x^2-6x+72=a{x}^3+b{x}^2+cx-4a{x}^2-4bx-4c\\ \Rightarrow x^3-7x^2-6x+72=a{x}^3-4a{x}^2+b{x}^2+cx-4bx-4c\\ \Rightarrow x^3-7x^2-6x+72=a{x}^3+\left(b-4a\right)x^2+x\left(c-4b\right)-4c\end{array}$

On comparing

We get

  $\begin{array}{l}a=1\\ b-4a=-7\\ c-4b=-6\\ -4c=72\end{array}$

  $\begin{array}{l}b-4a=-7\\ \Rightarrow b-4\left(1\right)=-7\\ \Rightarrow b-4=-7\\ \Rightarrow b=-7+4\\ \Rightarrow b=-3\end{array}$

  $\begin{array}{l}c-4b=-6\\ \Rightarrow c-4\left(-3\right)=-6\\ \Rightarrow c+12=-6\\ \Rightarrow c=-6-12\\ \Rightarrow c=-18\end{array}$

Therefore,

  $f\left(x\right)=x^3-7x^2-6x+72=\left(x-4\right)\left(x^2-3x-18\right)$

On simplifying

We get

  $\begin{array}{l}f\left(x\right)=\left(x-4\right)\left(x^2-3x-18\right)\\ \Rightarrow f\left(x\right)=\left(x-4\right)\left(x^2-6x+3x-18\right)\\ \Rightarrow f\left(x\right)=\left(x-4\right)\left(x\left(x-6\right)+3\left(x-6\right)\right)\\ \Rightarrow f\left(x\right)=\left(x-4\right)\left(x+3\right)\left(x-6\right)\end{array}$

Therefore,

Factored form of given expression is

  $f\left(x\right)=\left(x-4\right)\left(x+3\right)\left(x-6\right)$