Question

In the following exercises, factor completely.

$u^5+u^2$

Short answer

The factored form of the polynomial is$u^2\left(u+1\right)\left(u^2-u+1\right).$

Step-by-step solution

Step 1. Given Information.

The given polynomial is $u^5+u^2.$

Step 2. Factoring 

Let's find the GCF and factor out the GCF,

$u^2\left(u^3+1\right)$

In the parentheses, the polynomial is a binomial and it is the type of the sum of cubes.

To factor, the given polynomial completely we will use the sum of cubes formula.

So,

$$\begin{gathered} u^2\left(u^3+1\right) \\ =u^2\left\{\left(u+1\right)\left(u^2-u+1\right)\right\} \\ =u^2\left(u+1\right)\left(u^2-u+1\right) \end{gathered}$$

The polynomial is factored completely.

Thus, the factored form is $u^2\left(u+1\right)\left(u^2-u+1\right).$

Step 3. Check the answer 

To check the answer, expand the factored form we get,

$$\begin{gathered} u^5+u^2=u^2\left(u+1\right)\left(u^2-u+1\right) \\ {u}^5+u^2=u^3+u^2\left(u^2-u+1\right) \\ {u}^5+u^2=u^5-u^4+u^3+u^4-u^3+u^2 \\ {u}^5+u^2=u^5+u^2 \end{gathered}$$

L.H.S = R.H.S

Thus, the answer is right.