Factoring Basics
Learning Outcomes
- Identify and factor the greatest common factor of a polynomial.
- Factor a trinomial with leading coefficient 1.
- Factor by grouping.
Recall that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For example, 4 is the GCF of 16 and 20 because it is the largest number that divides evenly into both 16 and 20. The GCF of polynomials works the same way: 4x is the GCF of 16x and 20x2 because it is the largest polynomial that divides evenly into both 16x and 20x2.
Finding and factoring out a GCF from a polynomial is the first skill involved in factoring polynomials.
Factoring a GCF out of a polynomial
When factoring a polynomial expression, our first step is to check to see if each term contains a common factor. If so, we factor out the greatest amount we can from each term. To make it less challenging to find this GCF of the polynomial terms, first look for the GCF of the coefficients, and then look for the GCF of the variables.
A General Note: Greatest Common Factor
The greatest common factor (GCF) of a polynomial is the largest polynomial that divides evenly into each term of the polynomial.
To factor out a GCF from a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property "backwards" to rewrite the polynomial in a factored form. Recall that the distributive property of multiplication over addition states that a product of a number and a sum is the same as the sum of the products.
Distributive Property Forward and Backward
Forward: We distribute a over b+c.
a(b+c)=ab+ac.
Backward:
We factor a out of ab+ac.
ab+ac=a(b+c).
We have seen that we can distribute a factor over a sum or difference. Now we see that we can "undo" the distributive property with factoring.
Example
Factor
25b3+10b2.
Answer: Find the GCF.
25b3=5⋅5⋅b⋅b⋅b10b2=5⋅2⋅b⋅bGCF=5⋅b⋅b=5b2
Rewrite each term with the GCF as one factor.
25b3=5b2⋅5b10b2=5b2⋅2
Rewrite the polynomial using the factored terms in place of the original terms.
5b2(5b)+5b2(2)
Factor out the
5b2.
5b2(5b+2)
Answer
5b2(5b+2)
Analysis
The factored form of the polynomial
25b3+10b2 is
5b2(5b+2). You can check this by doing the multiplication.
5b2(5b+2)=25b3+10b2.
Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.
For example:
25b3+10b2=5(5b3+2b2)Factor out 5.=5b2(5b+2)Factor out b2.
Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.
In the following video we see two more examples of how to find and factor the GCF from binomials.
https://youtu.be/25_f_mVab_4
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.
Example: Factoring the Greatest Common Factor
Factor
6x3y3+45x2y2+21xy.
Answer:
First find the GCF of the expression. The GCF of 6,45, and 21 is 3. The GCF of x3,x2, and x is x. (Note that the GCF of a set of expressions of the form xn will always be the lowest exponent.) The GCF of y3,y2, and y is y. Combine these to find the GCF of the polynomial, 3xy.
Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that 3xy(2x2y2)=6x3y3,3xy(15xy)=45x2y2, and 3xy(7)=21xy.
Finally, write the factored expression as the product of the GCF and the sum of the terms we needed to multiply by.
(3xy)(2x2y2+15xy+7)
Analysis of the Solution
After factoring, we can check our work by multiplying. Use the distributive property to confirm that
(3xy)(2x2y2+15xy+7)=6x3y3+45x2y2+21xy.
The GCF may not always be a monomial. Here is an example of a GCF that is a binomial.
Try It
Factor
x(b2−a)+6(b2−a) by pulling out the GCF.
Answer:
(b2−a)(x+6)
[ohm_question]7888[/ohm_question]
Watch this video to see more examples of how to factor the GCF from a trinomial.
https://youtu.be/3f1RFTIw2Ng
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 x2+5x+6 has a GCF of 1, but it can be written as the product of the factors (x+2) and (x+3).
Trinomials of the form x2+bx+c can be factored by finding two numbers with a product of c and a sum of b. The trinomial x2+10x+16, for example, can be factored using the numbers 2 and 8 because the product of these numbers is 16 and their sum is 10. The trinomial can be rewritten as the product of (x+2) and (x+8).
A General Note: Factoring a Trinomial with Leading Coefficient 1
A trinomial of the form
x2+bx+c can be written in factored form as
(x+p)(x+q) where
pq=c and
p+q=b.
Q & A
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 form x2+bx+c, factor it
- List factors of c.
- Find p and q, a pair of factors of c with a sum of b.
- Write the factored expression (x+p)(x+q).
Example: Factoring a Trinomial with Leading Coefficient 1
Factor
x2+2x−15.
Answer:
We have a trinomial with leading coefficient 1,b=2, and c=−15. We need to find two numbers with a product of −15 and a sum of 2. In the table, we list factors until we find a pair with the desired sum.
Factors of −15 |
Sum of Factors |
1,−15 |
−14 |
−1,15 |
14 |
3,−5 |
−2 |
−3,5 |
2 |
Now that we have identified
p and
q as
−3 and
5, write the factored form as
(x−3)(x+5).
Analysis of the Solution
We can check our work by multiplying. Use FOIL to confirm that
(x−3)(x+5)=x2+2x−15.
Q & A
Does the order of the factors matter?
No. Multiplication is commutative, so the order of the factors does not matter.
Try It
Factor
x2−7x+6.
Answer:
(x−6)(x−1)
[ohm_question]7897[/ohm_question]
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 2x2+5x+3 can be rewritten as (2x+3)(x+1) using this process. We begin by rewriting the original expression as 2x2+2x+3x+3 and then factor each portion of the expression to obtain 2x(x+1)+3(x+1). We then pull out the GCF of (x+1) to find the factored expression.
A General Note: Factoring by Grouping
To factor a trinomial of the form
ax2+bx+c by grouping, we find two numbers with a product of
ac and a sum of
b. We use these numbers to divide the
x term 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 form ax2+bx+c, factor by grouping
- List factors of ac.
- Find p and q, a pair of factors of ac with a sum of b.
- Rewrite the original expression as ax2+px+qx+c.
- Pull out the GCF of ax2+px.
- Pull out the GCF of qx+c.
- Factor out the GCF of the expression.
Example: Factoring a Trinomial by Grouping
Factor
5x2+7x−6 by grouping.
Answer:
We have a trinomial with a=5,b=7, and c=−6. First, determine ac=−30. We need to find two numbers with a product of −30 and a sum of 7. In the table, we list factors until we find a pair with the desired sum.
Factors of −30 |
Sum of Factors |
1,−30 |
−29 |
−1,30 |
29 |
2,−15 |
−13 |
−2,15 |
13 |
3,−10 |
−7 |
−3,10 |
7 |
So
p=−3 and
q=10.
5x2−3x+10x−6x(5x−3)+2(5x−3)(5x−3)(x+2)Rewrite the original expression as ax2+px+qx+c.Factor out the GCF of each part.Factor out the GCF of the expression.
Analysis of the Solution
We can check our work by multiplying. Use FOIL to confirm that
(5x−3)(x+2)=5x2+7x−6.
Try It
Factor the following.
- 2x2+9x+9
- 6x2+x−1
Answer:
- (2x+3)(x+3)
- (3x−1)(2x+1)
[ohm_question]7908[/ohm_question]
In the next video we see another example of how to factor a trinomial by grouping.
[embed]https://youtu.be/agDaQ_cZnNc[/embed]Licenses & Attributions
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- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2.
- Example: Greatest Common Factor. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Factoring Trinomials by Grouping. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Question ID 7886, 7897, 7908. Authored by: Tyler Wallace. License: CC BY: Attribution. License terms: IMathAS Community License CC- BY + GPL.
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- College Algebra. Provided by: OpenStax Authored by: OpenStax College Algebra. Located at: https://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1/Preface. License: CC BY: Attribution.