Prerequisites

# Factoring Polynomials

### Learning Objectives

In this section students will:

- Factor the greatest common factor of a polynomial.
- Factor a trinomial.
- Factor by grouping.
- Factor a perfect square trinomial.
- Factor a difference of squares.
- Factor the sum and difference of cubes.
- Factor expressions using fractional or negative exponents.

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in (Figure).

The area of the entire region can be found using the formula for the area of a rectangle.

The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area ofunits^{2}. The other rectangular region has one side of lengthand one side of lengthgiving an area ofunits^{2}. So the region that must be subtracted has an area ofunits^{2}.

The area of the region that requires grass seed is found by subtractingunits^{2}. This area can also be expressed in factored form asunits^{2}. We can confirm that this is an equivalent expression by multiplying.

Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

### Factoring the Greatest Common Factor of a Polynomial

When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance,is the GCF ofandbecause it is the largest number that divides evenly into bothandThe GCF of polynomials works the same way:is the GCF ofandbecause it is the largest polynomial that divides evenly into bothand

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

### Greatest Common Factor

The greatest common factor (GCF) of polynomials is the largest polynomial that divides evenly into the polynomials.

### How To

**Given a polynomial expression, factor out the greatest common factor.
**

- Identify the GCF of the coefficients.
- Identify the GCF of the variables.
- Combine to find the GCF of the expression.
- Determine what the GCF needs to be multiplied by to obtain each term in the expression.
- Write the factored expression as the product of the GCF and the sum of the terms we need to multiply by.

### Factoring the Greatest Common Factor

Factor

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First, find the GCF of the expression. The GCF ofandisThe GCF ofandis(Note that the GCF of a set of expressions in the formwill always be the exponent of lowest degree.) And the GCF ofandisCombine these to find the GCF of the polynomial,

Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find thatand

Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.

#### Analysis

After factoring, we can check our work by multiplying. Use the distributive property to confirm that

### Try It

Factorby pulling out the GCF.

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### Factoring a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomialhas a GCF of 1, but it can be written as the product of the factorsand

Trinomials of the formcan be factored by finding two numbers with a product ofand a sum ofThe trinomialfor example, can be factored using the numbersandbecause the product of those numbers isand their sum isThe trinomial can be rewritten as the product ofand

### Factoring a Trinomial with Leading Coefficient 1

A trinomial of the formcan be written in factored form aswhereand

**Can every trinomial be factored as a product of binomials?**

*No. Some polynomials cannot be factored. These polynomials are said to be prime.*

### How To

**Given a trinomial in the formfactor it.**

- List factors of
- Findanda pair of factors ofwith a sum of
- Write the factored expression

### Factoring a Trinomial with Leading Coefficient 1

Factor

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We have a trinomial with leading coefficientandWe need to find two numbers with a product ofand a sum ofIn (Figure), we list factors until we find a pair with the desired sum.

Factors of | Sum of Factors |
---|---|

14 | |

2 |

Now that we have identifiedandasandwrite the factored form as[/hidden-answer]

#### Analysis

We can check our work by multiplying. Use FOIL to confirm that

**Does the order of the factors matter?**

*No. Multiplication is commutative, so the order of the factors does not matter.*

### Try It

Factor

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### Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the *x* term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomialcan be rewritten asusing this process. We begin by rewriting the original expression asand then factor each portion of the expression to obtainWe then pull out the GCF ofto find the factored expression.

### Factor by Grouping

To factor a trinomial in the formby grouping, we find two numbers with a product ofand a sum ofWe use these numbers to divide theterm into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

### How To

**Given a trinomial in the formfactor by grouping.**

- List factors of
- Findanda pair of factors ofwith a sum of
- Rewrite the original expression as
- Pull out the GCF of
- Pull out the GCF of
- Factor out the GCF of the expression.

### Factoring a Trinomial by Grouping

Factorby grouping.

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We have a trinomial withandFirst, determineWe need to find two numbers with a product ofand a sum ofIn (Figure), we list factors until we find a pair with the desired sum.

Factors of | Sum of Factors |
---|---|

29 | |

13 | |

7 |

Soand

#### Analysis

We can check our work by multiplying. Use FOIL to confirm that

Factor a.b.

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a.b.

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### Factoring a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

We can use this equation to factor any perfect square trinomial.

### Perfect Square Trinomials

A perfect square trinomial can be written as the square of a binomial:

### How To

**Given a perfect square trinomial, factor it into the square of a binomial.
**

- Confirm that the first and last term are perfect squares.
- Confirm that the middle term is twice the product of
- Write the factored form as

### Factoring a Perfect Square Trinomial

Factor

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Notice thatandare perfect squares becauseandThen check to see if the middle term is twice the product ofandThe middle term is, indeed, twice the product:Therefore, the trinomial is a perfect square trinomial and can be written as

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### Try It

Factor

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### Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

We can use this equation to factor any differences of squares.

### Differences of Squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

### How To

**Given a difference of squares, factor it into binomials.**

- Confirm that the first and last term are perfect squares.
- Write the factored form as

### Factoring a Difference of Squares

Factor

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Notice thatandare perfect squares becauseandThe polynomial represents a difference of squares and can be rewritten as

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### Try It

Factor

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**Is there a formula to factor the sum of squares?**

*No. A sum of squares cannot be factored.*

### Factoring the Sum and Difference of Cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: **S**ame **O**pposite **A**lways **P**ositive. For example, consider the following example.

The sign of the first 2 is the *same* as the sign betweenThe sign of theterm is *opposite* the sign betweenAnd the sign of the last term, 4, is *always positive*.

### Sum and Difference of Cubes

We can factor the sum of two cubes as

We can factor the difference of two cubes as

### How To

**Given a sum of cubes or difference of cubes, factor it.**

- Confirm that the first and last term are cubes,or
- For a sum of cubes, write the factored form asFor a difference of cubes, write the factored form as

### Factoring a Sum of Cubes

Factor

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#### Analysis

After writing the sum of cubes this way, we might think we should check to see if the trinomial portion can be factored further. However, the trinomial portion cannot be factored, so we do not need to check.

### Try It

Factor the sum of cubes:

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### Factoring a Difference of Cubes

Factor

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#### Analysis

Just as with the sum of cubes, we will not be able to further factor the trinomial portion.

### Try It

Factor the difference of cubes:

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### Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance,can be factored by pulling outand being rewritten as

### Factoring an Expression with Fractional or Negative Exponents

Factor

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Factor out the term with the lowest value of the exponent. In this case, that would be

### Try It

Factor

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Access these online resources for additional instruction and practice with factoring polynomials.

### Key Equations

difference of squares | |

perfect square trinomial | |

sum of cubes | |

difference of cubes |

- The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See (Figure).
- Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See (Figure).
- Trinomials can be factored using a process called factoring by grouping. See (Figure).
- Perfect square trinomials and the difference of squares are special products and can be factored using equations. See (Figure) and (Figure).
- The sum of cubes and the difference of cubes can be factored using equations. See (Figure) and (Figure).
- Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See (Figure).

#### Verbal

If the terms of a polynomial do not have a GCF, does that mean it is not factorable? Explain.

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The terms of a polynomial do not have to have a common factor for the entire polynomial to be factorable. For example,anddon’t have a common factor, but the whole polynomial is still factorable:

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A polynomial is factorable, but it is not a perfect square trinomial or a difference of two squares. Can you factor the polynomial without finding the GCF?

How do you factor by grouping?

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Divide theterm into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression.

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#### Algebraic

For the following exercises, find the greatest common factor.

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For the following exercises, factor by grouping.

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For the following exercises, factor the polynomial.

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For the following exercises, factor the polynomials.

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#### Real-World Applications

For the following exercises, consider this scenario:

Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city’s parks. The park is a rectangle with an area ofm^{2}, as shown in the figure below. The length and width of the park are perfect factors of the area.

Factor by grouping to find the length and width of the park.

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A statue is to be placed in the center of the park. The area of the base of the statue isFactor the area to find the lengths of the sides of the statue.

At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain isFactor the area to find the lengths of the sides of the fountain.

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For the following exercise, consider the following scenario:

A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure below. The flagpole will take up a square plot with areayd^{2}.

Find the length of the base of the flagpole by factoring.

#### Extensions

For the following exercises, factor the polynomials completely.

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### Glossary

- factor by grouping
- a method for factoring a trinomial in the formby dividing the
*x*term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression

- greatest common factor
- the largest polynomial that divides evenly into each polynomial