Question

How do you factor $f\left( x \right)={{x}^{3}}-12{{x}^{2}}+36x-32$ completely given that $\left( x-2 \right)$ is the factor?

Answer 1

Hint: Now we are given that $\left( x-2 \right)$ is the factor of the equation. Now we will try to write the given expression in the form $f\left( x \right)\left( x-2 \right)$ . Hence we will get a quadratic $f\left( x \right)$ . Now to factorize the quadratic expression we will split the middle term. Hence on simplifying we get the factors of the quadratic equation. Hence we have factors of all the equations.

Complete step by step solution:
No consider the given expression ${{x}^{3}}-12{{x}^{2}}+36x-32$
Now we know that $x-2$ is the factor of this equation.
We will try to write the expression in the form of $\left( x-2 \right)f\left( x \right)={{x}^{3}}-12{{x}^{2}}+36x-32$ .
Now let us first split $-12{{x}^{2}}=-2{{x}^{2}}-10{{x}^{2}}$ .
Hence we have ${{x}^{3}}-2{{x}^{2}}-10{{x}^{2}}+36x-32$ .
Now taking ${{x}^{2}}$ common from the first two terms we get,
$\Rightarrow {{x}^{2}}\left( x-2 \right)-10{{x}^{2}}+36x-32$
Now writing $36x=20x+16x$ we get the expression as,
$\Rightarrow {{x}^{2}}\left( x-2 \right)-10{{x}^{2}}+20x+16x-32$
Now taking -10 common from second and third term then we get,
$\Rightarrow {{x}^{2}}\left( x-2 \right)-10x\left( x-2 \right)+16x-32$ .
Now taking 16 common from last two terms we get,
$\begin{align}
  & \Rightarrow {{x}^{2}}\left( x-2 \right)-10x\left( x-2 \right)+16\left( x-2 \right) \\
 & \Rightarrow \left( x-2 \right)\left( {{x}^{2}}-10x+16 \right) \\
\end{align}$
Now we will try to factorize the quadratic equation.
To do so we will use the splitting the middle term method. Hence we get,
Hence we will split $-10x=-8x-2x$ .
$\Rightarrow \left( x-2 \right)\left( {{x}^{2}}-2x-8x+16 \right)$
Now taking x common from first two terms and -8 common from last two terms we get,
$\begin{align}
  & \Rightarrow \left( x-2 \right)x\left( x-2 \right)-8\left( x-2 \right) \\
 & \Rightarrow \left( x-2 \right)\left( x-8 \right)\left( x-2 \right) \\
\end{align}$
Hence we get the factors of the equation as $\left( x-2 \right)$ , $\left( x-8 \right)$ and $\left( x-2 \right)$ .

Note: Now note that to factorize the quadratic equation with splitting the middle terms method we will split the middle terms such that the product of two terms is multiplication of first term and last term. Hence in the given quadratic we split $-10x=-8x-2x$ as $\left( -8x \right)\left( -2x \right)=16{{x}^{2}}={{x}^{2}}\times 16$ .