# 6.8: Factoring Polynomials in Quadratic Form

**At Grade**Created by: CK-12

**Practice**Methods for Solving Quadratic Functions

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* 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.

Factor to Solve

"Factor to Solve" is a common method for solving quadratic equations accomplished by factoring a trinomial into two binomials and identifying the values of that make each binomial equal to zero.factored form

The factored form of a quadratic function is , where and are the roots of the function.Factoring

Factoring is the process of dividing a number or expression into a product of smaller numbers or expressions.Quadratic form

A polynomial in quadratic form looks like a trinomial or binomial and can be factored like a quadratic expression.quadratic function

A quadratic function is a function that can be written in the form , where , , and are real constants and .Roots

The roots of a function are the values of*x*that make

*y*equal to zero.

standard form

The standard form of a quadratic function is .Vertex form

The vertex form of a quadratic function is , where is the vertex of the parabola.Zeroes of a Polynomial

The zeroes of a polynomial are the values of that cause to be equal to zero.### Image Attributions

Here you'll how to factor and solve polynomials that are in “quadratic form.”

## Concept Nodes:

Factor to Solve

"Factor to Solve" is a common method for solving quadratic equations accomplished by factoring a trinomial into two binomials and identifying the values of that make each binomial equal to zero.factored form

The factored form of a quadratic function is , where and are the roots of the function.Factoring

Factoring is the process of dividing a number or expression into a product of smaller numbers or expressions.Quadratic form

A polynomial in quadratic form looks like a trinomial or binomial and can be factored like a quadratic expression.quadratic function

A quadratic function is a function that can be written in the form , where , , and are real constants and .Roots

The roots of a function are the values of*x*that make

*y*equal to zero.

standard form

The standard form of a quadratic function is .Vertex form

The vertex form of a quadratic function is , where is the vertex of the parabola.Zeroes of a Polynomial

The zeroes of a polynomial are the values of that cause to be equal to zero.