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

### Guidance

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.

#### Example A

Factor .

**
Solution:
**
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.

#### Example B

Factor .

**
Solution:
**
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.

#### Example C

Find all the real-number solutions of .

**
Solution:
**
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.

**
Intro Problem Revisit
**
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 .

### Guided Practice

Factor the following polynomials.

1.

2.

3. Find all the real-number solutions of .

#### Answers

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

2. Factor this polynomial like a difference of squares.

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

3. Pull out a from each term.

Set each factor equal to zero.

Notice the second factor will give imaginary solutions.

### Vocabulary

- Quadratic form
- When a polynomial looks a trinomial or binomial and can be factored like a quadratic equation.

### Practice

Factor the following quadratics completely.

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