Q. 413 - Intermediate Algebra

Question

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

$b^3-64$

Step-by-Step Solution

Answer

The factors of the given expression are $(b-4)(b^2+4b+16)$.

Step 1. Given and explanation

We have $b^3-64$.

We can see that this forms a pattern of difference of two perfect cubes.

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

Applying this, we will factorize the given expression.

Step 2. Factorizing the expression.

We have $b^3-64$.

This can be written as ${\left(b\right)}^3-{\left(4\right)}^3$.

Applying the property ${\left(a\right)}^3-{\left(b\right)}^3=(a-b)(a^2+ab+b^2)$, we get

${\left(b\right)}^3-{\left(4\right)}^3=(b-4)(b^2+4b+16)$

This cannot be further factorized, so the factors of the given expression are $(b-4)(b^2+4b+16)$.

Step 3. Check the solution.

We will check the solution by simply multiplying the resulting factors.

If we get the given equation as a product, our factorization is right.

So,

$$\begin{gathered} =(b-4)(b^2+4b+16) \\ =b^3+4b^2+16b-4b^2-16b-64 \\ =b^3-64 \end{gathered}$$

Thus our calculations are right.