9.6: Monomial Factors of Polynomials
Suppose that the height of a golf ball above the ground is represented by the expression
Guidance
We have been multiplying polynomials by using the Distributive Property, where all the terms in one polynomial must be multiplied by all terms in the other polynomial. In this Concept, you will start learning how to do this process using a different method called factoring.
Factoring: Take the factors that are common to all the terms in a polynomial. Then multiply the common factors by a parenthetical expression containing all the terms that are left over when you divide out the common factors.
Let’s look at the areas of the rectangles again:
Method 1: Find the areas of all the small rectangles and add them.
Blue rectangle
Orange rectangle
Red rectangle
Green rectangle
Purple rectangle
Total area
Method 2: Find the area of the big rectangle all at once.
The answers are the same no matter which method you use:
Finding the Greatest Common Monomial Factor
Once we get a polynomial in factored form, it is easier to solve the polynomial equation. But first, we need to learn how to factor. Factoring can take several steps because we want to factor completely until we cannot factor any more.
A common factor can be a number, a variable, or a combination of numbers and variables that appear in every term of the polynomial.
When a common factor is factored from a polynomial, you divide each term by the common factor. What is left over remains in parentheses.
Example A
Factor:

15x−25 
3a+9b+6
Solution:
1. We see that the factor of 5 divides evenly from all terms.
2. We see that the factor of 3 divides evenly from all terms.
Now we will use examples where different powers can be factored and there is more than one common factor.
Example B
Find the greatest common factor.
Solution:
Notice that the factor
Let’s rewrite:
Factor
Example C
Find the greatest common factor.
Solution:
The common factor is
When we factor
Vocabulary
Common factor: A common factor can be a number, a variable, or a combination of numbers and variables that appear in every term of the polynomial.
Guided Practice
Find the greatest common factor.
Solution:
First, look at the coefficients to see if they share any common factors. They do: 4.
Next, look for the lowest power of each variable, because that is the most you can factor out. The lowest power of
This means we can factor out
Practice
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK12 Basic Algebra: Polynomial Equations in Factored Form (9:29)
Factor the common factor from the following polynomials.

36a2+9a3−6a7 
yx3y2+12x+16y 
3x3−21x 
5x6+15x4 
4x3+10x2−2x 
−10x6+12x5−4x4 
12xy+24xy2+36xy3 
5a3−7a 
45y12+30y10 
16xy2z+4x3y
Mixed Review
 Rewrite in standard form:
−4x+11x4−6x7+1−3x2 . State the polynomial's degree and leading coefficient.  Simplify
(9a2−8a+11a3)−(3a2+14a5−12a)+(9−3a5−13a) .  Multiply
13a3 by(36a4+6) .  Melissa made a trail mix by combining
x ounces of a 40% cashew mixture with \begin{align*}y\end{align*} ounces of a 30% cashew mixture. The result is 12 ounces of cashews. Write the equation to represent this situation.
 Graph using its intercepts.
 Give three possible combinations to make this sentence true.
 Explain how to use mental math to simplify 8(12.99).
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common factor
A common factor can be a number, a variable, or a combination of numbers and variables that appear in every term of the polynomial.Greatest Common Factor
The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.Monomial
A monomial is an expression made up of only one term.Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Image Attributions
Here you'll learn how to identify the greatest common factor for the terms in a polynomial so that you can simplify by factoring.