Question

How do you factor the polynomial $4{x^3} + 32$?

Answer 1

Hint: We are given an equation and we have to find its factor. Firstly we will take the factor out of the equation. Then after the equation is formed then apply the algebraic identity. Here according to the equation we will apply the identity. Identity which involves the sum of the cubes of two numbers.
Formula used:
$A^3+B^3= \left(A+B \right) \left(A^2-AB+B^2 \right)$

Complete step by step solution:
Step1: we are given an equation $4{x^3} + 32$.
Firstly we will factor out $4$ from the equation as$4$is common in both the terms.
$ \Rightarrow 4\left( {{x^3} + 8} \right)$
Step2: Now if we recognize that $\left( {{x^3} + 8} \right)$ is the sum of two cubes as ${x^3}$ is the cube of $x$ and $8$ is of $2$. $x \times x \times x$ And $2 \times 2 \times 2$ with signs + between them we will apply the algebraic identities for $\left( {{x^3} + 8} \right)$.
Using the identity:
$A^3+B^3= \left(A+B \right) \left(A^2-AB+B^2 \right)$
Here ${A^3} = {x^3};{B^3} = 2^3$
On substituting the values in the identity we will get the value:
$ \Rightarrow \left( {{x^3} + 2^3} \right) = (x + 2)({x^2} - 2x + 4)$
So the factors will be $4\left( {x + 2} \right)\left( {{x^2} - 2x + 4} \right)$
Step3: Hence the factors are $4\left( {x + 2} \right)\left( {{x^2} - 2x + 4} \right)$

Note: In such types of questions students mainly don't get an approach how to solve it. In such types of algebraic expressions mainly first we have to factor out a number from an expression then the expression forms in a way so that it can be factorized and identity can be applied. There are many algebraic identities apart from this which we have used in this question we have to just apply the identity correctly. Many times students forget the identity as there are many identities so they mix match them and apply them wrong. So before solving such questions revise the identities so the chances of getting questions wrong becomes very less. The main approach to solve these questions is to form a question in such a way and then apply identity. Sometimes we have to equate the equation to zero then the algebraic identity gets applied which factorizes the whole expression. Different identities can be applied whether it is of square or cube depending upon the equation.
Commit to memory:
$A^3-B^3= \left(A+B \right) \left(A^2-AB+B^2 \right)$