Question

How do you factor ${{x}^{3}}-4{{x}^{2}}-25x+100$?

Answer 1

Hint: Now to factor the given expression we will first take ${{x}^{2}}$ common from the first two terms and $-25$ common from the last two terms. Now we will again take the common terms together and hence obtain a simplified expression. Now we will use the formula ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ and hence factorize the expression.

Complete step by step solution:
Now the given expression is a cubic expression.
We cannot find the roots or factors of a cubic expression directly.
Now consider the expression ${{x}^{3}}-4{{x}^{2}}-25x+100$
To factorize the expression we will first try to simplify and find one factor of the expression.
Now first we will simplify the terms of the expression by grouping common terms.
Hence let us take ${{x}^{2}}$ common from the first two terms and -25 common from last two terms.
Hence we get,
$\Rightarrow {{x}^{3}}-4{{x}^{2}}-25x+100={{x}^{2}}\left( x-4 \right)-25\left( x-4 \right)$ .
Now taking $x-4$ common we get the given expression as
$\Rightarrow {{x}^{3}}-4{{x}^{2}}-25x+100=\left( x-4 \right)\left( {{x}^{2}}-25 \right)$
Now we have one factor of the given expression which is $\left( x-4 \right)$ Now to find the other factors we will factorize the quadratic expression.
Now we know that ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ .
Hence using this we can write $\left( {{x}^{2}}-25 \right)=\left( x-5 \right)\left( x+5 \right)$ .
Hence we get,
$\Rightarrow \left( x-4 \right)\left( x-5 \right)\left( x+5 \right)$
Hence the factors of the given expression are $\left( x-4 \right),\left( x-5 \right)$ and $\left( x+5 \right)$ .

Note: Now note that such simplification is not always possible in the expression. Hence we can substitute different values of x in the expression to find a root of expression. Now let us say we have $\alpha $ as the root of the given expression. Then we have \[x-\alpha \] is the factor of the given expression. Now we will divide the given expression by $x-\alpha $ and hence get a quadratic expression and we will factorize the quadratic expression.