Learning Objectives
In this section students will:
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).
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 [latex]\,A={s}^{2}={4}^{2}=16\,[/latex] units2. The other rectangular region has one side of length [latex]\,10x-8\,[/latex] and one side of length [latex]\,4,[/latex] giving an area of [latex]\,A=lw=4\left(10x-8\right)=40x-32\,[/latex] units2. So the region that must be subtracted has an area of [latex]\,2\left(16\right)+40x-32=40x\,[/latex] units2.
The area of the region that requires grass seed is found by subtracting [latex]\,60{x}^{2}-40x\,[/latex] units2. This area can also be expressed in factored form as [latex]\,20x\left(3x-2\right)\,[/latex] units2. We can confirm that this is an equivalent expression by multiplying.
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.
5.2.1 – 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, [latex]\,4\,[/latex] is the GCF of [latex]\,16\,[/latex] and [latex]\,20\,[/latex] because it is the largest number that divides evenly into both [latex]\,16\,[/latex] and [latex]\,20\,[/latex] The GCF of polynomials works the same way: [latex]\,4x\,[/latex] is the GCF of [latex]\,16x\,[/latex] and [latex]\,20{x}^{2}\,[/latex] because it is the largest polynomial that divides evenly into both [latex]\,16x\,[/latex] and [latex]\,20{x}^{2}.[/latex]
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 polynomials is the largest polynomial that divides evenly into the polynomials.
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 1 – Factoring the Greatest Common Factor
Factor [latex]\,6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy.[/latex]
First, find the GCF of the expression. The GCF of [latex]\,6,45,[/latex] and [latex]\,21\,[/latex] is [latex]\,3.\,[/latex] The GCF of [latex]\,{x}^{3},{x}^{2},[/latex] and [latex]\,x\,[/latex] is [latex]\,x.\,[/latex] (Note that the GCF of a set of expressions in the form [latex]\,{x}^{n}\,[/latex] will always be the exponent of lowest degree.) And the GCF of [latex]\,{y}^{3},{y}^{2},[/latex] and [latex]\,y\,[/latex] is [latex]\,y.\,[/latex] Combine these to find the GCF of the polynomial, [latex]\,3xy.[/latex]
Next, determine what the GCF needs to be multiplied by to obtain each term of the polynomial. We find that [latex]$$\,3xy\left(2{x}^{2}{y}^{2}\right)=6{x}^{3}{y}^{3},$$[/latex] [latex]$$3xy\left(15xy\right)=45{x}^{2}{y}^{2},$$[/latex] and [latex]$$\,3xy\left(7\right)=21xy.$$[/latex]
Finally, 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 [latex]\,\left(3xy\right)\left(2{x}^{2}{y}^{2}+15xy+7\right)=6{x}^{3}{y}^{3}+45{x}^{2}{y}^{2}+21xy.[/latex]
Try It
Factor [latex]\,x\left({b}^{2}-a\right)+6\left({b}^{2}-a\right)\,[/latex] by pulling out the GCF.
Show answer
[latex]\left({b}^{2}-a\right)\left(x+6\right)[/latex]
5.2.2 – 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 [latex]\,{x}^{2}+5x+6\,[/latex] has a GCF of 1, but it can be written as the product of the factors [latex]\,\left(x+2\right)\,[/latex] and [latex]\,\left(x+3\right).[/latex]
Trinomials of the form [latex]\,{x}^{2}+bx+c\,[/latex] can be factored by finding two numbers with a product of [latex]c\,[/latex] and a sum of [latex]\,b.\,[/latex] The trinomial [latex]\,{x}^{2}+10x+16,[/latex] for example, can be factored using the numbers [latex]\,2\,[/latex] and [latex]\,8\,[/latex] because the product of those numbers is [latex]\,16\,[/latex] and their sum is [latex]\,10.\,[/latex] The trinomial can be rewritten as the product of [latex]\,\left(x+2\right)\,[/latex] and [latex]\,\left(x+8\right).[/latex]
Factoring a Trinomial with Leading Coefficient 1
A trinomial of the form [latex]\,{x}^{2}+bx+c\,[/latex] can be written in factored form as [latex]\,\left(x+p\right)\left(x+q\right)\,[/latex] where [latex]\,pq=c\,[/latex] and [latex]\,p+q=b.[/latex]
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 [latex]\,{x}^{2}+bx+c,[/latex] factor it.
- List factors of [latex]\,c.[/latex]
- Find [latex]\,p\,[/latex] and [latex]\,q,[/latex] a pair of factors of [latex]\,c\,[/latex] with a sum of [latex]\,b.[/latex]
- Write the factored expression [latex]\,\left(x+p\right)\left(x+q\right).[/latex]
We have a trinomial with leading coefficient [latex]\,1,b=2,[/latex] and [latex]\,c=-15.\,[/latex] We need to find two numbers with a product of [latex]\,-15\,[/latex] and a sum of [latex]\,2.\,[/latex] In (Figure), we list factors until we find a pair with the desired sum.
| Factors of [latex]\,-15[/latex] | Sum of Factors |
|---|---|
| [latex]1,-15[/latex] | [latex]-14[/latex] |
| [latex]-1,15[/latex] | 14 |
| [latex]3,-5[/latex] | [latex]-2[/latex] |
| [latex]-3,5[/latex] | 2 |
Now that we have identified [latex]\,p\,[/latex] and [latex]\,q\,[/latex] as [latex]\,-3\,[/latex] and [latex]\,5,[/latex] write the factored form as [latex]\,\left(x-3\right)\left(x+5\right).[/latex]
Analysis
We can check our work by multiplying. Use FOIL to confirm that [latex]\,\left(x-3\right)\left(x+5\right)={x}^{2}+2x-15.[/latex]
Does the order of the factors matter?
No. Multiplication is commutative, so the order of the factors does not matter.
Try It
Factor [latex]\,{x}^{2}-7x+6.[/latex]
Show answer
[latex]\left(x-6\right)\left(x-1\right)[/latex]
Access these online resources for additional instruction and practice with factoring polynomials.
Key Equations
| difference of squares | [latex]{a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)[/latex] |
| perfect square trinomial | [latex]{a}^{2}+2ab+{b}^{2}={\left(a+b\right)}^{2}[/latex] |
| sum of cubes | [latex]{a}^{3}+{b}^{3}=\left(a+b\right)\left({a}^{2}-ab+{b}^{2}\right)[/latex] |
| difference of cubes | [latex]{a}^{3}-{b}^{3}=\left(a-b\right)\left({a}^{2}+ab+{b}^{2}\right)[/latex] |
- The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See (Example 1).
- Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See (Example 2).
Glossary
- factor by grouping
- a method for factoring a trinomial in the form [latex]\,a{x}^{2}+bx+c\,[/latex] 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
- greatest common factor
- the largest polynomial that divides evenly into each polynomial