The volume of a rectangular prism is . What are the lengths of the prism's sides?

### Factoring Polynomials in Quadratic Form

The last type of factorable polynomial are those that are in quadratic form. **Quadratic form** is when a polynomial looks like a trinomial or binomial and can be factored like a quadratic. One example is when a polynomial is in the form . Another possibility is something similar to the difference of squares, . This can be factored to or . Always keep in mind that the greatest common factors should be factored out first.

#### Factor

Factor .

This particular polynomial is factorable. Let’s use the method we learned in the *Factoring When the Leading Coefficient Doesn't Equal 1* concept. First, . The factors of -30 that add up to -1 are -6 and 5. Expand the middle term and then use factoring by grouping.

Both of the factors are not factorable, so we are done.

Factor .

Treat this polynomial equation like a difference of squares.

Now, we can factor using the difference of squares a second time.

cannot be factored because it is a sum of squares. This will have imaginary solutions.

Find all the real-number solutions of .

First, pull out the GCF among the three terms.

Factor what is inside the parenthesis like a quadratic equation. and the factors of -18 that add up to -17 are -18 and 1. Expand the middle term and then use factoring by grouping.

Factor further and solve for where possible. is not factorable.

### Examples

#### Example 1

Earlier, you were asked what are the lengths of the prism's sides.

To find the lengths of the prism's sides, we need to factor .

First, pull out the GCF among the three terms.

Factor what is inside the parenthesis like a quadratic equation. and the factors of -6 that add up to -5 are -6 and 1.

Therefore, the lengths of the rectangular prism's sides are , , and .

Factor the following polynomials.

#### Example 2

and the factors of 24 that add up to 14 are 12 and 2.

#### Example 3

Factor this polynomial like a difference of squares.

6 and 5 are not square numbers, so this cannot be factored further.

#### Example 4

Find all the real-number solutions of .

Pull out a from each term.

Set each factor equal to zero.

Notice the second factor will give imaginary solutions.

### Review

Factor the following quadratics completely.

Find all the real-number solutions to the polynomials below.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.8.