Question

How do you factor by grouping ${x^3} + 4{x^2} + 8x + 32$?

Answer 1

Hint: In this question, we want to find factors of the given expression. For that, we will use the grouping method. The first group has the first two terms, and the second group has another two terms. Then, take out the greatest common factor from each group. Now, take out the common term.

Complete step-by-step answer:
In this question, we want to factor the expression ${x^3} + 4{x^2} + 8x + 32$ by grouping the terms.
Let us make a pair of terms of the given expression.
$ \Rightarrow {x^3} + 4{x^2} + 8x + 32$
Now, the first group has the first two terms, and the second group has another two terms.
Here, the first group will be ${x^3} + 4{x^2}$ , and the second group will be $8x + 32$
So, let us write them into brackets.
$ \Rightarrow \left( {{x^3} + 4{x^2}} \right) + \left( {8x + 32} \right)$
Now, take out the greatest common factor from each group.
In the first group, ${x^2}$ is the greatest common factor.
And in the second group, 8 is the greatest common factor.
Therefore, we will get:
$ \Rightarrow {x^2}\left( {x + 4} \right) + 8\left( {x + 4} \right)$
Now, take out the common term to get factors.
Here, $x + 4$ is the common term in the above expression.
So, we will get:
$ \Rightarrow \left( {x + 4} \right)\left( {{x^2} + 8} \right)$

Hence, the factors of the given expression ${x^3} + 4{x^2} + 8x + 32$ are $x + 4$ and ${x^2} + 8$.

Note:
 One important thing is, we can always check our work by multiplying out factors back together, and check that we have got back the original answer.
To check our factorization, multiplication goes like this:
$ \Rightarrow \left( {x + 4} \right)\left( {{x^2} + 8} \right)$
Break the first bracket and multiply it with another bracket.
So,
$ \Rightarrow x\left( {{x^2} + 8} \right) + 4\left( {{x^2} + 8} \right)$
Let us apply multiplication to remove brackets.
$ \Rightarrow {x^3} + 8x + 4{x^2} + 32$
Let us rewrite the above expression, we will get:
$ \Rightarrow {x^3} + 4{x^2} + 8x + 32$
Therefore, by performing the reverse process we will get our given expression back.
Hence, it is proved that our answer is right.