Question

In the following exercises, factor completely.

$10m^4-6250$

Short answer

The factored form of the polynomial is $10\left(m-5\right)(m+5)(m^2+25).$

Step-by-step solution

Step 1. Given Information.

The given polynomial is $10m^4-6250.$

Step 2. Factoring

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

$10\left(m^4-625\right)$

In the parentheses, the polynomial is a binomial and it is the type of the difference of squares.

To factor, the given polynomial completely writes it as a product of conjugates.

So,

$$\begin{gathered} 10\left(m^4-625\right) \\ =10\left\{{\left(m^2\right)}^2-{\left(25\right)}^2\right\} \\ =10\left\{\left(m^2-25\right)\left(m^2+25\right)\right\} \\ =10\left\{{\left(m\right)}^2-{\left(5\right)}^2\right\}\left(m^2+25\right) \\ =10\left(m-5\right)(m+5)(m^2+25) \end{gathered}$$

The polynomial is factored completely.

Thus, the factored form is $10\left(m-5\right)(m+5)(m^2+25).$

Step 3. Check the answer 

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

$$\begin{gathered} 10m^4-6250=10\left(m-5\right)\left(m+5\right)\left(m^2+25\right) \\ 10m^4-6250=10m-50\left(m+5\right)\left(m^2+25\right) \\ 10m^4-6250=10m^2+50m-50m-250\left(m^2+25\right) \\ 10m^4-6250=10m^4+250m^2+50m^3+1250m-50m^3 \\ -1250m-250m^2-6250 \\ 10m^4-6250=10m^4-6250 \end{gathered}$$

Thus, the answer is right.