Question

How do you factorize the given polynomial $4{{x}^{3}}-8{{x}^{2}}-12x$?

Answer 1

Hint: We start solving the problem by writing all the terms in the polynomial as a multiplication of two terms. We then take the common terms present in the polynomial after multiplication to problems. We then perform the subtraction and multiplication operations and then take the common factors from the obtained result to get the required answer for the factorization of the given polynomial.

Complete step by step answer:
According to the problem, we are asked to factorize the given polynomial $4{{x}^{3}}-8{{x}^{2}}-12x$.
We have given the polynomial $4{{x}^{3}}-8{{x}^{2}}-12x$.
$\Rightarrow 4{{x}^{3}}-8{{x}^{2}}-12x=\left( 4{{x}^{2}}\times x \right)-\left( 8x\times x \right)-\left( 12\times x \right)$ ---(1).
From equation (1), we can see that each term has a common factor x. So, let us take out the common factor to find the other factors of the given polynomial.
$\Rightarrow 4{{x}^{3}}-8{{x}^{2}}-12x=x\times \left( 4{{x}^{2}}-8x-12 \right)$.
$\Rightarrow 4{{x}^{3}}-8{{x}^{2}}-12x=x\times \left( 4{{x}^{2}}-12x+4x-12 \right)$.
$\Rightarrow 4{{x}^{3}}-8{{x}^{2}}-12x=x\times \left( \left( 4x\times x \right)-\left( 4x\times 3 \right)+\left( 4\times x \right)-\left( 4\times 3 \right) \right)$ ---(2).
Let us take the common terms present in equation (2).
$\Rightarrow 4{{x}^{3}}-8{{x}^{2}}-12x=x\times \left( 4x\times \left( x-3 \right)+4\times \left( x-3 \right) \right)$ ---(3).
From equation (3), we can see that each term has a common factor $\left( x-3 \right)$. So, let us take out the common factor to find the other factors of the given polynomial.
$\Rightarrow 4{{x}^{3}}-8{{x}^{2}}-12x=x\times \left( \left( 4x+4 \right)\left( x-3 \right) \right)$.
$\Rightarrow 4{{x}^{3}}-8{{x}^{2}}-12x=x\left( 4x+4 \right)\left( x-3 \right)$.
So, we have found the factors of the given polynomial $4{{x}^{3}}-8{{x}^{2}}-12x$ as $x$, $\left( 4x+4 \right)$, $\left( x-3 \right)$.

$\therefore $ The factorization of the given polynomial $4{{x}^{3}}-8{{x}^{2}}-12x$ is $x\left( 4x+4 \right)\left( x-3 \right)$.

Note: Whenever we get this type of problems, we first try to find the common multiples (factors) in all the terms present in the polynomial. We should keep in mind that we need to get the same polynomial after multiplying the factors obtained from factorization. We can also report the answer as $4x\left( x+1 \right)\left( x-3 \right)$ or $x\left( x+1 \right)\left( 4x-12 \right)$ which are also same but the difference is that different factor is multiplied with 4 in different results. Similarly, we can expect problems to find the factorization of the polynomial $4{{x}^{3}}-3{{x}^{2}}-x$.