## Key Concepts

- Factor a Polynomial Model
- Find the Greatest Common Factor
- Factor out the Greatest Common Factor

### Factoring Polynomials

#### 1. Find the Greatest Common Factor

**What is the Greatest Common Factor (GCF) of the terms of 12x**^{5}** + 8x**^{4}** – 6x**^{3}**?**

**Step 1. **Write the prime factorization of the coefficient for each term to determine if there is a greater common factor other than 1.

**Step 2. **Determine the greatest common factor for the variables of each term.

The greatest common factor of the terms 12x^{5} + 8x^{4} – 6x^{3}** ^{ }**is 2x

^{3}.

#### 2. Factor out the Greatest Common Factor

**Why is it helpful to factor out the GCF from a polynomial?**

Consider the polynomial -12x^{3} + 18x+2 – 27x.

**Step 1. **Find the GCF of the terms of the polynomial, if there is one. Because the first term is negative, it is helpful to factor out -1.

The greatest common factor is -3x.

**Step 2. **Factor the GCF out of each term of the polynomial.

-3x (4x^{2} – 6x + 9)

Factoring out the greatest common factor results in a polynomial with smaller coefficients and/or smaller exponents of the variable(s). This makes it easier to analyze the polynomial or factor it further.

### Application

**Alani is in charge of marketing for a travel company. She is designing a brochure that will have 6 photos. The photos can be arranged on the page in a number of ways. **

1. **What is the total area of the photos?**

First, find the area of each type of photo.

Area = **Area of square photos** + **area of narrower photos**

**= 2(x**^{2}**) + 4(1x)**** **

= 2x^{2} + 4x

The total area of the photos is 2x^{2} + 4x square in.

2. **Find a rectangular arrangement for the photos. What factored expression represents the area of the arrangement?**

Try placing the photos in one row.

The factored form that represents the area of the arrangement is x(2x + 4).

3. **Factor out the GCF from the polynomial. What does the GCF represent in this situation?**

The GCF of 2x^{2} and 4x is 2x. So, you can rewrite the expression as 2x(x + 2).

The GCF represents the height of one possible arrangement of the photos.

4. **Which of these two arrangements is a practical use of the space on a page of the brochure?**

The arrangement based on the GCF is more practical because the arrangement with the photos in one line will likely be too wide for a page.

### Questions

**Question 1**

Find the GCF of each term of a polynomial.

**1. 15x ^{2} + 18 **

**Solution:**

GCF of the coefficients:

15 = 3×5

18 = 2×3×3

Here GCF is 3.

GCF of the variables:* *The only common factor between x^{2} and x^{0} is x^{0}, i.e., 1.

So, **GCF is 3.**

**2. -18y ^{4} + 6y^{3} + 24y^{2} **

**Solution:**

GCF of the coefficients:

-18 = (-1) ×2×3×3

6 = 2×3

24 = 2×2×2×3

Here GCF is 2 × 3 = 6

GCF of the variables:

Y^{4} = y×y×y×y

Y^{3} = y×y×y

Y^{2} = y×y

Here GCF is y × y i.e., y^{2}.

So, **GCF is 6y ^{2}.**

**Question 2**

Factor out the GCF from each polynomial.

**1. x ^{3} + 5x^{2} – 22x **

**Solution:**

x^{3} = x × x × x

5x^{2} = 5 × x × x

-22x = (-2) × 11 × x

Here, GCF is x.

**x (x**^{2}** + 5x – 22) **

**2. -16y ^{6} + 28y^{4} – 20y^{3} **

**Solution:**

-16y^{6} = (-1) × 2 × 2 × 2 × 2 × y × y × y × y × y × y

28y^{4} = (-1) × 2 × 2 × (-7) × y × y × y × y

-20y^{3} = (-1) × 2 × 2 × 5 × y × y × y

Here, GCF is (-1) × 2 × 2 × y × y × y i.e. -4y^{3}.

**-4y**^{3}** (4y**^{3}** – 7y + 5) **

**Question 3**

In the last example mentioned in the previous section, suppose the dimensions of the narrower photos were increased to 2 in. by x in. What expression would represent the new arrangement based on the GCF?

**Solution:**

Area of square photos = 2(x^{2})

Area of narrower photos = 4(2x) = 8x

Total area of the photos = 2x^{2} + 8x

2x^{2} = 2 × x × x

8x = 2 × 2 × 2 × x

So, GCF is 2 × x, i.e., 2x.

2x^{2} + 8x = 2x (x + 4)

Area of the new arrangement based on the GCF =** 2x (x + 4) square in. **

### Key Concepts Covered

#### Words

Determine if a polynomial can be factored. If the polynomial can be factored, find the greatest common factor of the terms and factor it out.

## Exercise

- How is factoring a polynomial similar to factoring integers?
- Why does the GCF of the variables of a polynomial have the least exponent of any variable term in the polynomial?
- What is the greatest common factor of two polynomials that do not appear to have any common factors?
- Andrew factored 3x^2y – 6xy
^{2}+ 3xy as 3xy(x – 2y). Describe and correct his error. - What term and 12x^2y have a GCF of 4xy? Write an expression that shows the monomial factored out of the polynomial.
- Find the GCF of the terms of the following polynomials:

- 8x
^{6}+ 32x^{3} - 15x + 27
- 7x
^{4}– x^{3} - 6ab
^{2}+ 8ab – b - 86ab
^{2}+ 64b – 34a

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