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 [link].
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 ofunits2. The other rectangular region has one side of lengthand one side of lengthgiving an area ofunits2. So the region that must be subtracted has an area ofunits2.
The area of the region that requires grass seed is found by subtractingunits2. This area can also be expressed in factored form asunits2. 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.
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
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
Factorby pulling out the GCF.
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.
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
We have a trinomial with leading coefficientandWe need to find two numbers with a product ofand a sum ofIn [link], 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
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.
Factor
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.
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.
We have a trinomial withandFirst, determineWe need to find two numbers with a product ofand a sum ofIn [link], 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.
a.b.
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:
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
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
Factor
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.
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
Notice thatandare perfect squares becauseandThe polynomial represents a difference of squares and can be rewritten as
Factor
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: Same Opposite Always Positive. 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
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
Notice thatandare cubes becauseRewrite the sum of cubes as
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.
Factor the sum of cubes:
Factoring a Difference of Cubes
Factor
Notice thatandare cubes becauseandWrite the difference of cubes as
Analysis
Just as with the sum of cubes, we will not be able to further factor the trinomial portion.
Factor the difference of cubes:
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
Factor out the term with the lowest value of the exponent. In this case, that would be
Factor
Access these online resources for additional instruction and practice with factoring polynomials.
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 [link].
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 [link].
Trinomials can be factored using a process called factoring by grouping. See [link].
Perfect square trinomials and the difference of squares are special products and can be factored using equations. See [link] and [link].
The sum of cubes and the difference of cubes can be factored using equations. See [link] and [link].
Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See [link].
Verbal
If the terms of a polynomial do not have a GCF, does that mean it is not factorable? Explain.
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:
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?
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.
Algebraic
For the following exercises, find the greatest common factor.
For the following exercises, factor by grouping.
For the following exercises, factor the polynomial.
For the following exercises, factor the polynomials.
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 ofm2, 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.
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.
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 areayd2.
Find the length of the base of the flagpole by factoring.
Extensions
For the following exercises, factor the polynomials completely.
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