Question

How do you factor $3{{x}^{3}}+{{x}^{2}}-75x-25$?

Answer 1

Hint: Now to factor the expression we will first group the first two terms and the last two terms together. Now from the first bracket we will take ${{x}^{2}}$ common and from the last bracket we will take 25 common. Now we will simplify the expression and use the formula ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ to further simplify the expression. Hence we have the factors of the given expression.

Complete step by step solution:
Now consider the given expression $3{{x}^{3}}+{{x}^{2}}-75x-25$
Now we want to find the factors of the given equation.
Let us first understand the concept of factors.
Now we know that factors of numbers are nothing but the numbers which divide the number given.
Similarly factors of expressions are nothing but expressions which divide the given expression.
Now let us take an example to understand this.
Take the expression ${{x}^{2}}+2x$ .
Now we can take x common from the expression Hence we get,
$\Rightarrow x\left( x+2 \right)$ .
Now we can see that the expression will not leave a remainder when divided by x or x + 2.
Hence we can say that x and x + 2 are the factors of the given expression.
Now consider the given equation $3{{x}^{3}}+{{x}^{2}}-75x-25$
We will try to find the factors by grouping the terms.
Consider $\left( 3{{x}^{3}}+{{x}^{2}} \right)-\left( 75x+25 \right)$
Now taking ${{x}^{2}}$ common from the first bracket and $25$ common from the second bracket we get,
$\Rightarrow {{x}^{2}}\left( 3x+1 \right)-25\left( 3x+1 \right)$
Now again taking the term $\left( 3x+1 \right)$ common we get,
$\begin{align}
  & \Rightarrow \left( {{x}^{2}}-25 \right)\left( 3x+1 \right) \\
 & \Rightarrow \left( {{x}^{2}}-{{5}^{2}} \right)\left( 3x+1 \right) \\
\end{align}$
Now we know that ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ . Hence we get,
$\Rightarrow \left( x+5 \right)\left( x-5 \right)\left( 3x+1 \right)$

Hence the factors of the given expression are $\left( x+5 \right)$, $\left( x-5 \right)$ and $\left( 3x+1 \right)$.

Note: Now note that the factors of the expression can easily help us to find the roots of the expression. Now if $ax+b$ is the factor of the expression then on solving the linear equation $ax+b=0$ we can get the root of the equation. Similarly if $\alpha $ is the root of the equation then we have $\left( x-\alpha \right)$ as the factor of the equation.