1.6: Factoring Polynomials
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In this section, you 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 1.
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 of
The area of the region that requires grass seed is found by subtracting
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,
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.
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
- Answer
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First, find the GCF of the expression. The GCF of 6, 45, and 21 is 3. The GCF of
, and is . (Note that the GCF of a set of expressions in the form will always be the exponent of lowest degree.) And the GCF of , and is . Combine 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 that
, andFinally, 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
Factor
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 polynomial
Trinomials of the form
A trinomial of the form
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 form
- List factors of
- Find
and a pair of factors of with a sum of - Write the factored expression
Factoring a Trinomial with Leading Coefficient 1
Factor
- Answer
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We have a trinomial with leading coefficient
and We need to find two numbers with a product of and a sum of In the table below, we list factors until we find a pair with the desired sum.Factors of Sum of Factors 14 2 Now that we have identified
and as and write 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 trinomial
To factor a trinomial in the form
Given a trinomial in the form
- List factors of
- Find
and a pair of factors of with 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
Factor
- Answer
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We have a trinomial with
and First, determine We need to find two numbers with a product of and a sum of In the table below, we list factors until we find a pair with the desired sum.Factors of Sum of Factors 29 13 7 So
and
Analysis
We can check our work by multiplying. Use FOIL to confirm that
Factor
- ⓐ
- ⓑ
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.
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
- Answer
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Notice that
and are perfect squares because and Then check to see if the middle term is twice the product of and The 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.
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
- Answer
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Notice that
and are perfect squares because and The 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 difference 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 between
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 as
For a difference of cubes, write the factored form as
Factoring a Sum of Cubes
Factor
- Answer
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Notice that
and are cubes because Rewrite 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
- Answer
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Notice that
and are cubes because and Write 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,
Factoring an Expression with Fractional or Negative Exponents
Factor
- Answer
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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.
1.5 Section Exercises
Verbal
If the terms of a polynomial do not have a GCF, does that mean it is not factorable? Explain.
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?
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 of
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 is
At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is
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 area
Find the length of the base of the flagpole by factoring.