Section7.1Factoring out the Common Factor
¶In earlier sections, we learned how to multiply polynomials, such as when you start with \((x+2)(x+3)\) and obtain \(x^2+5x+6\text{.}\) This chapter, starting with this section, is about the opposite processâfactoring. For example, starting with \(x^2+5x+6\) and obtaining \((x+2)(x+3)\text{.}\) We will start with the simplest kind of factoring: for example starting with \(x^2+2x\) and obtaining \(x(x+2)\text{.}\)
Subsection7.1.1Motivation for Factoring
When you write \(x^2+2x\text{,}\) you have a mathematical expression built with two termsâtwo parts that are added together. When you write \(x(x+2)\text{,}\) you have a mathematical expression built with two factorsâtwo parts that are multiplied together. Factoring is useful, because sometimes (but not always) having your expression written as parts that are multiplied together makes it easy to simplify the expression.
You've seen this with fractions. To simplify \(\frac{15}{35}\text{,}\) breaking down the numerator and denominator into factors is useful: \(\frac{3\cdot5}{7\cdot5}\text{.}\) Now you can see that the factors of \(5\) cancel.
There are a few other reasons to appreciate the value of factoring that will float to the surface in this chapter and beyond.
Subsection7.1.2Identifying the Greatest Common Factor
The most basic technique for factoring involves recognizing the greatest common factor between two expressions, which is the largest factor that goes in evenly to both expressions. For example, the greatest common factor between \(6\) and \(8\) is \(2\) since \(2\) goes in nicely into both \(6\) and \(8\) and no larger number would divide both \(6\) and \(8\) nicely.
Similarly, the greatest common factor between \(4x\) and \(3x^2\) is \(x\text{.}\) If you write \(4x\) as a product of its factors, you have \(2\cdot 2 \cdot x\text{.}\) And if you fully factor \(3x^2\text{,}\) you have \(3\cdot x\cdot x\text{.}\) The only factor they have in common is \(x\text{,}\) so that is the greatest common factor. No larger expression goes in nicely to both expressions.
Example7.1.2Finding the Greatest Common Factor
What is the common factor between \(6x^2\) and \(70x\text{?}\) Break down each of these into its factors:
\begin{align*} 6x^2 \amp =2\cdot3\cdot x\cdot x \amp 70x \amp =2\cdot5\cdot7\cdot x\\ \end{align*}And identify the common factors:
\begin{align*} 6x^2 \amp =\attention{2}\cdot3\cdot \attention{x}\cdot x \amp 70x \amp =\attention{2}\cdot5\cdot7\cdot \attention{x} \end{align*}With \(2\) and \(x\) in common, the greatest common factor is \(2x\text{.}\)
Let's try a few more examples.
Exercise7.1.3
Subsection7.1.3Factoring Out the Greatest Common Factor
We have learned the <<xref without ref, first/last, or provisional attribute (check spelling)>>: \(a(b+c)=ab+ac\text{.}\) Perhaps you have thought of this as a way to âdistributeâ the number \(a\) to each of \(b\) and \(c\text{.}\) In this section, we will use the distributive property in the opposite way. If you have an expression \(ab+ac\text{,}\) it is equal to \(a(b+c)\text{.}\) In that example, we factored out \(a\text{,}\) which is the common factor between \(ab\) and \(ac\text{.}\)
The following steps use the Distributive Property to factor out the greatest common factor between two or more terms.
Algorithm7.1.4Factoring Out the Greatest Common FactorâFilling in the Blank
Steps
Example
Identify the common factor in all terms.
Write the common factor outside a pair of parentheses with the appropriate addition or subtraction signs inside.
For each term from the original expression, what would you multiply the greatest common factor by to result in that term? Write your answer in the parentheses.
To factor \(12x^2+15x\text{:}\)
The common factor between \(12x^2\) and \(15x\) is \(3x\text{.}\)
\(3x(\phantom{4x}+\phantom{5})\)
\(3x(4x+5)\)
Let's look at a few examples.
Example7.1.5
Factor the polynomial \(3x^3+3x^2-9\text{.}\)
We identify the common factor as \(3\text{,}\) because \(3\) is the only common factor between \(3x^3\text{,}\) \(3x^2\) and \(9\text{.}\)
-
We write:
\begin{equation*} 3x^3+3x^2-9=3(\phantom{x^2}+\phantom{x^2}-\phantom{3})\text{.} \end{equation*} -
We ask the question â\(3\) times what gives \(3x^3\text{?}\)â The answer is \(x^3\text{.}\) Now we have:
\begin{equation*} 3x^3+3x^2-9=3(x^3+\phantom{x^2}-\phantom{3})\text{.} \end{equation*}We ask the question â\(3\) times what gives \(3x^2\text{?}\)â The answer is \(x^2\text{.}\) Now we have:
\begin{equation*} 3x^3+3x^2-9=3(x^3+x^2-\phantom{3})\text{.} \end{equation*}We ask the question â\(3\) times what gives \(9\text{?}\)â The answer is \(3\text{.}\) Now we have:
\begin{equation*} 3x^3+3x^2-9=3(x^3+x^2-3)\text{.} \end{equation*}
To check that this is correct, multiplying through \(3(x^3+x^2-3)\) should give the original expression \(3x^3+3x^2-9\text{.}\) We check this, and it does.
Exercise7.1.6
Subsection7.1.4Visualizing With Rectangles
In Section 6.3, we learned one way to multiply polynomials using rectangle diagrams. Similarly, we can factor a polynomial with a rectangle diagram.
Algorithm7.1.7Factoring Out the Greatest Common FactorâUsing Rectangles
Steps
Put the terms into adjacent rectangles. Think of these as labeling the areas of each rectangle.
Identify the common factor, and mark the height of the overall rectangle with it.
Mark the base of each rectangle based on each rectangle's area and height.
Since the overall rectangle's area equals its base times its height, the height is one factor, and the sum of the widths is another factor.
Example
We will factor \(12x^2+15x\text{,}\) the same polynomial from the example in Algorithm 7.1.4, so that you may compare the two styles.
| \(12x^2\) | \(15x\) |
| \(3x\) | \(12x^2\) | \(15x\) |
| \(4x\) | \(5\) | |
| \(3x\) | \(12x^2\) | \(15x\) |
So \(12x^2+15x\) factors as \(3x(4x+5)\text{.}\)
Subsection7.1.5More Examples of Factoring out the Common Factor
Previous examples did not cover every nuance with factoring out the greatest common factor. Here are a few more factoring examples that attempt to do so.
Example7.1.8
Factor \(-35m^5+5m^4-10m^3\text{.}\)
First, we identify the common factor. The number \(5\) is the greatest common factor of the three coefficients (which were \(-35\text{,}\) \(5\text{,}\) and \(-10\)) and also \(m^3\) is the largest expression that divides \(m^5\text{,}\) \(m^4\text{,}\) and \(m^3\text{.}\) Therefore the greatest common factor is \(5m^3\text{.}\)
In this example, the leading term is a negative number. When this happens, we will make it common practice to take that negative as part of the greatest common factor. So we will proceed by factoring out \(-5m^3\text{.}\) Note the sign changes.
\begin{align*} -35m^5\highlight{{}+{}}5m^4\highlight{{}-{}}10m^3\amp=-5m^3(\phantom{7m^2}\highlight{{}-{}}\phantom{m}\highlight{{}+{}}\phantom{2})\\ \amp=-5m^3(7m^2-\phantom{m}+\phantom{2})\\ \amp=-5m^3(7m^2-m+\phantom{2})\\ \amp=-5m^3(7m^2-m+2) \end{align*}Example7.1.9
Factor \(14-7n^2+28n^4-21n\text{.}\)
Notice that the terms are not in a standard order, with powers of \(n\) decreasing as you read left to right. It is usually a best practice to rearrange the terms into the standard order first. The only exception is sometimes with multivariable expressions.
\begin{equation*} 14-7n^2+28n^4-21n=28n^4-7n^2-21n+14\text{.} \end{equation*}Next, the number \(7\) divides all of the numerical coefficients. Separately, no power of \(n\) is part of the greatest common factor because the \(14\) term has no \(n\) factors. So the greatest common factor is just \(7\text{.}\) So we proceed:
\begin{align*} 14-7n^2+28n^4-21n\amp=28n^4-7n^2-21n+14\\ \amp=7\mathopen{}\left(4n^4-n^2-3n+2\right)\mathclose{} \end{align*}Example7.1.10
Factor \(24ab^2+16a^2b^3-12a^3b^2\text{.}\)
There are two variables in this polynomial, but that does not change the factoring strategy. The greatest numerical factor between the three terms is \(4\text{.}\) The variable \(a\) divides all three terms, and \(b^2\) divides all three terms. So we have:
\begin{align*} 24ab^2+16a^2b^3-12a^3b^2 \amp=4ab^2\mathopen{}\left(6+4ab-3a^2\right)\mathclose{} \end{align*}Example7.1.11
Factor \(4m^2n-3xy\text{.}\)
There are no common factors in those two terms (unless you want to count \(1\) or \(-1\text{,}\) but we do not count these for the purposes of identifying a greatest common factor). In this situation we can say the polynomial is prime or irreducible, and leave it as it is.
Example7.1.12
Factor \(-x^3+2x+18\text{.}\)
There are no common factors in those three terms, and it would be correct to state that this polynomial is prime or irreducible. However, since its leading coefficient is negative, it may be wise to factor out a negative sign. So, it could be factored as \(-\mathopen{}\left(x^3-2x-18\right)\mathclose{}\text{.}\) Note that every term is negated as the leading negative sign is extracted.
Subsection7.1.6Exercises
Identifying Common Factors
Factoring out the Common Factor