Learning Objectives
- Review the concept of greatest common factor
- Factor a Polynomial
Factor Tree for 54
In the section on the zero product principle, we showed that using the techniques for solving equations that we learned for linear equations did not work to solve
[latex]t\left(5-t\right)=0[/latex]
But because the equation was written as the product of two terms, we could use the zero product principle. What if we are given a polynomial equation that is not written as a product of two terms, such as this one [latex]2y^2+4y=0[/latex]? We can use a technique called factoring, where we try to find factors that can be divided into each term of the polynomial so it can be rewritten as a product.
In this section we will explore how to find common factors from the terms of a polynomial, and rewrite it as a product. This technique will help us solve polynomial equations in the next section.
Greatest Common Factor
When we studied fractions, we learned that 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 [latex]20x^2[/latex] because it is the largest polynomial that divides evenly into both 16x and [latex]20x^2[/latex].
A General Note: Greatest Common Factor
The greatest common factor (GCF) of a group of given polynomials is the largest polynomial that divides evenly into the polynomials.
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 and 5 and 1 and 20. To factor a number is to rewrite it as a product. [latex]20=4\cdot5[/latex]. In algebra, we use the word factor as both a noun – something being multiplied – and as a verb – rewriting a sum or difference as a product.
To factor a polynomial, you rewrite it as a product. Any integer can be written as the product of factors, and we can apply this technique to monomials or polynomials. Factoring is very helpful in simplifying and solving equations using polynomials.
Prime Numbers
A prime factor is similar to a prime number—it has only itself and 1 as factors. The process of breaking a number down into its prime factors is called prime factorization.
To get acquainted with the idea of factoring, let’s first find the greatest common factor (GCF) of two whole numbers. The GCF of two numbers is the greatest number that is a factor of both of the numbers. Take the numbers 50 and 30.
[latex]\begin{array}{l}50=10\cdot5\\30=10\cdot3\end{array}[/latex]
Their greatest common factor is 10, since 10 is the greatest factor that both numbers have in common.
To find the GCF of greater numbers, you can factor each number to find their prime factors, identify the prime factors they have in common, and then multiply those together.
Example
Find the greatest common factor of 210 and 168.
Because the GCF is the product of the prime factors that these numbers have in common, you know that it is a factor of both numbers. (If you want to test this, go ahead and divide both 210 and 168 by 42—they are both evenly divisible by this number!)
The video that follows show another example of finding the greatest common factor of two whole numbers.
Finding the greatest common factor in a set of monomials is not very different from finding the GCF of two whole numbers. The method remains the same: factor each monomial independently, look for common factors, and then multiply them to get the GCF.
Example
Find the greatest common factor of [latex]25b^{3}[/latex] and [latex]10b^{2}[/latex].
The monomials have the factors 5, b, and b in common, which means their greatest common factor is [latex]5\cdot{b}\cdot{b}[/latex], or simply [latex]5b^{2}[/latex].
The video that follows gives another example of finding the greatest common factor of two monomials with only one variable.
Example
Find the greatest common factor of [latex]81c^{3}d[/latex] and [latex]45c^{2}d^{2}[/latex].
The video that follows shows another example of finding the greatest common factor of two monomials with more than one variable.
Factor a Polynomial
One of these things is not like the others.
Before we solve polynomial equations, we will practice finding the greatest common factor of a polynomial. If you can find common factors for each term of a polynomial, then you can factor it, and solving will be easier.
To help you practice finding common factors, identify factors that the terms of the polynomial have in common in the table below.
| Polynomial | Terms | Common Factors |
|---|---|---|
| [latex]6x+9[/latex] | 6x and 9 | 3 is a factor of 6x and 9 |
| [latex]a^{2}–2a[/latex] | [latex]a^{2}[/latex] and [latex]−2a[/latex] | a is a factor of [latex]a^{2}[/latex] and [latex]−2a[/latex] |
| [latex]4c^{3}+4c[/latex] | [latex]4c^{3}[/latex] and [latex]4c[/latex] | 4 and c are factors of [latex]4c^{3}[/latex] and 4c |
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 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: Product of a number and a sum: [latex]a\left(b+c\right)=a\cdot{b}+a\cdot{c}[/latex]. You can say that “[latex]a[/latex] is being distributed over [latex]b+c[/latex].”
Backward: Sum of the products: [latex]a\cdot{b}+a\cdot{c}=a\left(b+c\right)[/latex]. 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 [latex]25b^{3}+10b^{2}[/latex].
The factored form of the polynomial [latex]25b^{3}+10b^{2}[/latex] is [latex]5b^{2}\left(5b+2\right)[/latex]. You can check this by doing the multiplication. [latex]5b^{2}\left(5b+2\right)=25b^{3}+10b^{2}[/latex].
Note that if you do not factor the greatest common factor at first, you can continue factoring, rather than start all over.
For example:
[latex]\begin{array}{l}25b^{3}+10b^{2}=5\left(5b^{3}+2b^{2}\right)\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }5.\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=5b^{2}\left(5b+2\right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{Factor out }b^{2}.\end{array}[/latex]
Notice that you arrive at the same simplified form whether you factor out the GCF immediately or if you pull out factors individually.
Example
Factor [latex]81c^{3}d+45c^{2}d^{2}[/latex].
The following video provides two more examples of finding the greatest common factor of a binomial
Summary
A whole number, monomial, or polynomial can be expressed as a product of factors. You can use some of the same logic that you apply to factoring integers to factoring polynomials. To factor a polynomial, first identify the greatest common factor of the terms, and then apply the distributive property to rewrite the expression. Once a polynomial in [latex]a\cdot{b}+a\cdot{c}[/latex] form has been rewritten as [latex]a\left(b+c\right)[/latex], where a is the GCF, the polynomial is in factored form.