Question

How do you factor completely $2{{x}^{3}}+2{{x}^{2}}-12x$?

Answer 1

Hint: In this question, we have to find the factors of the given quadratic equation. We will first take out the term $2x$ common from the entire term and then we will solve it by splitting the middle term to get the factors for the same. We start solving this problem by finding two numbers such that the product of the two numbers is equal to the product of the coefficient of ${{x}^{2}}$ and the constant. Also, the sum of these two numbers is the coefficient of $x$. Then, after finding these $2$ numbers, we split the middle term as the sum of those two numbers and simplify to get the required solution.

Complete step by step solution:
We have the quadratic equation as:
$\Rightarrow 2{{x}^{3}}+2{{x}^{2}}-12x\to (1)$
On taking the term $2x$ common from all the terms, we get:
$\Rightarrow 2x\left( {{x}^{2}}+x-6 \right)$
Now the term ${{x}^{2}}+x-6$ is in the form of a quadratic equation therefore we will solve it by splitting the middle term.
As we know, the general form of quadratic equation is $a{{x}^{2}}+bx+c\to (2)$
On comparing equations $(1)$ and $(2)$, we get:
$a=1$
$b=1$
$c=-6$
To factorize, we have to find two numbers $m$ and $n$ such that $m+n=b$ and $m\times n=a\times c$.
Therefore, the product should be $-6$ and the sum should be $1$
We see that if $m=3$ and $n=-2$ , then we get $m+n=1$ and $m\times n=-6$
So, we will split the middle term as the addition of $3$ and $-2$. On substituting, we get:
$\Rightarrow 2x\left( {{x}^{2}}+3x-2x-6 \right)$
Now on taking the common terms, we get:
\[\Rightarrow 2x\left( x\left( x+3 \right)-2\left( x+3 \right) \right)\]
Now since the term $\left( x+3 \right)$is common in both the terms, we can take it out as common and write the equation as:
\[\Rightarrow 2x\left( x+3 \right)\left( x-2 \right)\], which is the required factorized form of the equation.

Note:
The alternate method to do this sum is by first finding a root of the expression by hit and trial method and then performing long division. The root of the equation is the value of $x$ which when substituted gives the value of the expression as $0$.
Note that when $x=2$, the equation becomes:
$\Rightarrow 2{{\left( 2 \right)}^{3}}+2{{\left( 2 \right)}^{2}}-12\left( 2 \right)$
On simplifying, we get:
$\Rightarrow 16+8-24=0$, therefore $x=2$is a root of the equation which implies $\left( x-2 \right)$ is a factor of the expression.
Now using this factor when division is performed, the other two factors $x$ and $\left( x+3 \right)$will be found.