# 9.9: Factor Polynomials Using Special Products

**Basic**Created by: CK-12

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**Practice**Factor Polynomials Using Special Products

What if the area of a square playground were 10,000 square feet? Instead of taking the square root of the area to find the length of one of the playground's sides, you could set up the equation

### Guidance

When we learned how to multiply binomials, we talked about two special products: the Sum and Difference Formula and the Square of a Binomial Formula. In this Concept, we will learn how to recognize and factor these special products.

**Factoring the Difference of Two Squares**

We use the Sum and Difference Formula to factor a difference of two squares. A difference of two squares can be a quadratic polynomial in this form:

In these problems, the key is figuring out what the

#### Example A

*Factor the difference of squares.*

(a)

(b)

**Solution:**

(a) Rewrite

We substitute the values of

The answer is

(b) Rewrite

**Factoring Perfect Square Trinomials**

A **perfect square trinomial** has the form:

The **factored form** of a perfect square trinomial has the form:

In these problems, the key is figuring out what the

#### Example B

*Factor x2+8x+16.*

**Solution:**

Check that the first term and the last term are perfect squares.

Check that the middle term is twice the product of the square roots of the first and the last terms. This is true also since we can rewrite them.

This means we can factor

#### Example C

*Factor*

**Solution:**

Rewrite

We notice that this is a perfect square trinomial and we can factor it as

**Solving Polynomial Equations Involving Special Products**

We have learned how to factor quadratic polynomials that are helpful in solving polynomial equations like

**Step 1: Rewrite** the equation in standard form such that: Polynomial expression = 0.

**Step 2: Factor** the polynomial completely.

**Step 3:** Use the Zero Product Property to set **each factor equal to zero.**

**Step 4: Solve** each equation from step 3.

**Step 5: Check** your answers by substituting your solutions into the original equation.

### Video Review

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### Guided Practice

*Solve the following polynomial equations.*

**Solution:** No need to rewrite because it is already in the correct form.

**Factor:** We write 6 as a product of the following numbers:

**Set each factor equal to zero:**

**Solve:**

**Check:** Substitute each solution back into the original equation.

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Factoring Special Products (10:08)

Factor the following perfect square trinomials.

x2+8x+16 x2−18x+81 −x2+24x−144 x2+14x+49 4x2−4x+1 25x2+60x+36 4x2−12xy+9y2 x4+22x2+121

Factor the following differences of squares.

x2−4 x2−36 −x2+100 x2−400 9x2−4 25x2−49 −36x2+25 16x2−81y2

Solve the following quadratic equations using factoring.

x2−11x+30=0 x2+4x=21 x2+49=14x x2−64=0 x2−24x+144=0 4x2−25=0 x2+26x=−169 −x2−16x−60=0

**Mixed Review**

- Find the value for
k that creates an infinite number of solutions to the system{3x+7y=1kx−14y=−2 . - A restaurant has two kinds of rice, three choices of mein, and four kinds of sauce. How many plate combinations can be created if you choose one of each?
- Graph
y−5=13(x+4) . Identify its slope. - $600 was deposited into an account earning 8% interest compounded annually.
- Write the exponential model to represent this situation.
- How much money will be in the account after six years?

- Divide:
489÷−315 . - Identify an integer than is even and not a natural number.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.9.

difference of two squares

The**has the form .**

*difference of two squares*Difference of Squares

A difference of squares is a quadratic equation in the form .Perfect Square Trinomial

A perfect square trinomial is a quadratic expression of the form (which can be rewritten as ) or (which can be rewritten as ).Quadratic form

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

## Description

## Learning Objectives

Here you'll learn how to factor quadratic polynomials by using special products.

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## Date Created:

Feb 24, 2012## Last Modified:

Oct 01, 2015## Vocabulary

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