The Greatest Common Factor
Learning Outcomes
- Identify the greatest common factor of a polynomial
Factors are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: 2 and 10 are factors of 20, as are 4,5,1,20. To factor a number is to rewrite it as a product. 20=4⋅5 or 20=1⋅20. In algebra, we use the word factor as both a noun – something being multiplied – and as a verb – the action of rewriting a sum or difference as a product. Factoring is very helpful in simplifying expressions and solving equations involving polynomials.
The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance, 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.
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 a group of given polynomials is the largest polynomial that divides evenly into the polynomials.
Example
Find the greatest common factor of
25b3 and
10b2.
Answer:
25b3=5⋅5⋅b⋅b⋅b10b2=5⋅2⋅b⋅bGCF=5⋅b⋅bGCF=5b2
In the example above, the monomials have the factors 5, b, and b in common, which means their greatest common factor is 5⋅b⋅b, or simply 5b2.
The video that follows gives an example of finding the greatest common factor of two monomials with only one variable.
https://youtu.be/EhkVBXRBC2s
Sometimes you may encounter a polynomial with more than one variable, so it is important to check whether both variables are part of the GCF. In the next example, we find the GCF of two terms which both contain two variables.
Example
Find the greatest common factor of
81c3d and
45c2d2.
Answer:
81c3d=3⋅3⋅3⋅3⋅c⋅c⋅c⋅d45c2d2=3⋅3⋅5⋅c⋅c⋅d⋅dGCF=3⋅3⋅c⋅c⋅dGCF=9c2d
The video that follows shows another example of finding the greatest common factor of two monomials with more than one variable.
https://youtu.be/GfJvoIO3gKQ
Now that you have practiced identifying the GCF of terms with one and two variables, we can apply this idea to factoring the GCF out of a polynomial. Notice that the instructions are now "Factor" instead of "Find the greatest common factor."
To factor a polynomial, first identify the greatest common factor of the terms. You can then use the distributive property to rewrite the polynomial in 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: Product of a number and a sum:
a(b+c)=a⋅b+a⋅c. You can say that “
a is being distributed over
b+c.”
Backward: Sum of the products:
a⋅b+a⋅c=a(b+c). Here you can say that “
a is being factored out.”
We first learned that we could distribute a factor over a sum or difference, now we are learning that we can "undo" the distributive property with factoring.
Example
Factor
25b3+10b2.
Answer:
Find the GCF. From a previous example, you found the GCF of 25b3 and 10b2 to be 5b2.
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)
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 show two more examples of how to find and factor the GCF from binomials.
https://youtu.be/25_f_mVab_4
We will show one last example of finding the GCF of a polynomial with several terms and two variables. No matter how large the polynomial, you can use the same technique described below to factor out its GCF.
How To: Given a polynomial expression, factor out the greatest common factor
- Identify the GCF of the coefficients.
- Identify the GCF of the variables.
- Write together 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
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 in the form xn will always be the exponent of lowest degree.) And the GCF of y3,y2, and y is y. Put these together 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)
After factoring, we can check our work by multiplying. Use the distributive property to confirm that (3xy)(2x2y2+15xy+7)=6x3y3+45x2y2+21xy.
In the following video, you will see two more example of how to find and factor our the greatest common factor of a polynomial.
https://youtu.be/3f1RFTIw2NgLicenses & Attributions
CC licensed content, Original
CC licensed content, Shared previously
- Ex 1: Identify GCF and Factor a Binomial. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.
- Unit 12: Factoring, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Ex 2: Identify GCF and Factor a Trinomial. Authored by: James Sousa (Mathispower4u.com) . License: CC BY: Attribution.