Question

Factorize: $2{{x}^{3}}-3{{x}^{2}}-3x+2$

Answer 1

Hint: We will factor the given polynomial by grouping as many times as the requirement. We will split some terms accordingly. Then, we will group the polynomial and find the common terms. Then we will simplify the polynomial by taking the common factors out.

Complete step-by-step answer:
Let us consider the given polynomial, $2{{x}^{3}}-3{{x}^{2}}-3x+2$
We are asked to factorize the above polynomial. So, for that, we will split some terms accordingly. Thus, we will be able to group the polynomials.
Now, we will split the terms $-3{{x}^{2}}$ and $-3x$ into the sum of two terms.
We already know that $-3{{x}^{2}}={{x}^{2}}-4{{x}^{2}}$ and $-3x=-x-2x$
So, now, when we substitute them in the given polynomial, we will get $2{{x}^{3}}+{{x}^{2}}-x-4{{x}^{2}}-2x+2$
In the above form of the polynomial, we can see that there are terms we can take out.
So, we can take $x$ from the first three terms out and $-2$ from the last three terms.
As a result, we will get $x\left( 2{{x}^{2}}+x-1 \right)-2\left( 2{{x}^{2}}+x-1 \right)$
There is a common term $\left( 2{{x}^{2}}+x-1 \right)$ in the above polynomial which we can take out to get $\left( x-2 \right)\left( 2{{x}^{2}}+x-1 \right)$
Now, it is clear that $x-2$ is a factor of the given polynomial and thus $x=2$ is a root of the polynomial.
Now, we can find the other factors by factorize the polynomial $\left( 2{{x}^{2}}+x-1 \right)$
We will do grouping again.
We need to split the term $\left( 2{{x}^{2}}+2x-x-1 \right)$
We will take $2x$ from the first two terms out and $-1$ from the last two terms.
We will get $2x\left( x+1 \right)-1\left( x+1 \right)$
Now, we can take the term $x+1$ out to get $\left( 2x-1 \right)\left( x+1 \right)$
Now, we can write the given polynomial as $\left( x-2 \right)\left( 2x-1 \right)\left( x+1 \right)$
So, we will get the roots $x=2,\,x=\dfrac{1}{2}$ and $x=-1$ of the polynomial from its factors $x-2,\,2x-1$ and $x+1.$
Hence the factorization of the given polynomial is $\left( 2x-1 \right)\left( x-2 \right)\left( x+1 \right).$

Note: As we already know, we can use the quadratic formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ for a quadratic equation \[f\left( x \right)=a{{x}^{2}}+bx+c\] to find the solution of the equation. So, we can use this formula to find the solution of \[2{{x}^{2}}+x-1\] and thereby the factors of the given polynomial.