Q. 419 - Intermediate Algebra

Question

Recognize and Use the Appropriate Method to Factor a Polynomial Completely.

$10m^4-6250$.

Step-by-Step Solution

Answer

The factors of the given equation are $10(m^2+25)(m+5)(m-5)$.

Step 1. Given and explanation.

We have $10m^4-6250$.

We will first take out the common factors from the equation.

The we will get to see a pattern of difference of two perfect squares, i.e., ${\left(a\right)}^2-{\left(b\right)}^2=(a+b)(a-b)$.

By applying this, we will get the factors of the equation.

Step 2. Taking the common factor out and simplifying.

We have $10m^4-6250$.

Taking $10$ common gives us,

$10(m^4-625)$.

We know that ${\left(m^2\right)}^2=m^4$ and
${\left(25\right)}^2=625$.

Step 3. Finding factors and simplifying.

We have $10(m^4-625)$.

We know that they are perfect squares. So applying the property of difference of two perfect squares, we get

$=10(m^2+25)(m^2-25)$

We can further apply the property in $(m^2-25)$ as ${\left(m\right)}^2=m^2$ and
${\left(5\right)}^2=25$.

So by applying he property we get,

$=10(m^2+25)(m+5)(m-5)$

This cannot be further factored so they are the factors of the expression.

Step 4. 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,

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

Thus our calculations are right.