Q. 414 - Intermediate Algebra

Question

Recognize and use the appropriate method to factor a polynomial completely in the following exercises.

$m^3+125$.

Step-by-Step Solution

Answer

The factors of the given equation are  $(m+5)(m^2-5m+25)$.

Step 1. Given and explanation

We have $m^3+125$.

If we analyze, we get to see the identity of ${\left(a\right)}^3+{\left(b\right)}^3$.

We know that ${\left(a\right)}^3+{\left(b\right)}^3=(a+b)(a^2-ab+b^2)$.

So by applying this property and simplifying the equation, we will get the factors.

Step 2. Simplifying the equation.

We have $m^3+125$.

This can be written as ${\left(m\right)}^3+{\left(5\right)}^3$.

So now applying the property of sum of perfect cubes, it gives us,

${\left(m\right)}^3+{\left(5\right)}^3=(m+5)(m^2-5m+25)$.

This cannot be further factorized so they are the factors of the given equation. 

Step 3. Check the solution.

We will check the solution by simply multiplying the factors. If we get the given equation as the product, our answer is right.

So,

$$\begin{gathered} =(m+5)(m^2-5m+25) \\ =m^3-5m^2+25m+5m^2-25m+125 \\ =m^3+125 \end{gathered}$$

Thus our calculations are right.