Question

How do you factor ${{x}^{3}}-4{{x}^{2}}+4x-16$?

Answer 1

To factorize the given expression we will first group common terms. Hence we take ${{x}^{2}}$ common from the first two terms and 4 common from the last two terms. Now we will simplify the equation. Now we will find the roots of the obtained quadratic expression and hence find the factors. Hence we have the factors of the given expression.

Complete step by step solution:
Now the given polynomial is a cubic polynomial in x.
Now to factorize the polynomial we will first have to simplify the expression. To simplify the given expression we will group common terms from the expression.
Now let us simplify the expression by taking ${{x}^{2}}$ common from the first two terms and 4 common from the last two terms.
Hence we get the given expression as ${{x}^{2}}\left( x-4 \right)+4\left( x-4 \right)$
Now we will take $x-4$ common from the whole expression.
Hence we get the expression as $\left( x-4 \right)\left( {{x}^{2}}+4 \right)$
Now consider the quadratic expression ${{x}^{2}}+4$ .
Now we will find the roots of the expression.
To find the roots of the expression consider the equation ${{x}^{2}}+4=0$
Taking 4 from LHS to RHS we get, ${{x}^{2}}=-4$
Taking square root in the above equation we get,
$\Rightarrow x=\pm 2i$
Hence the factors of the quadratic expression are $\left( x+2i \right)$ and $\left( x-2i \right)$ .
Hence the factors of the given expression are $\left( x-4 \right)\left( x-2i \right)\left( x+2i \right)$

Note: Now note that the roots of quadratic equation of the form $a{{x}^{2}}+bx+c=0$ can be given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence we can get the roots of the equation and hence we can easily write the factors of the given expression. Note that in the above quadratic we have b as 0.