Factor Polynomial Expressions
By factoring polynomial expressions, we can see the structure of that expression differently and identify roots and multiplicities of those roots. While we will not cover every possible type of factoring in this section, we will identify some of the most common. In the next target where we solve polynomial equations and find roots of polynomial functions, we will use this factoring to help us.
As we talked about when factoring quadratic expressions (polynomials of degree \(2\)), we first want to always look for a greatest common factor (GCF) and factor that out if there is one. We will incorporate this into many of the examples below.
Fourth Degree Trinomial
If we look at the expression \(x^4-5x^2-6\), hopefully you will notice that it has a similar structure to \(x^2-5x-6\). Just like that expression can be factored into \(\left(x-6\right)\left(x+1\right)\), we can factor \(x^4-5x^2-6\) into \(\left(x^2-6\right)\left(x^2+1\right)\).
Factor the following expressions.
1. \(3x^4+2x^2-5\)
Solution:
\(\begin{align}&\left(3x^2+5\right)\left(x^2-1\right)\ & \ \ &\text{1) Factor the trinomial like you would factor}\ ax^2+bx+c.\\
&\left(3x^2+5\right)\left(x+1\right)\left(x-1\right)\ & \ \ &\text{2) Factor}\ \left(x^2-1\right)\ \text{ with the difference of squares pattern.}\end{align}\)
2. \(x^5-8x^3+12x\)
Solution:
\(\begin{align}&x(x^4-8x^2+12)\ & \ \ &\text{1) Factor out the GCF.}\\
&x\left(x^2-6\right)\left(x^2-2\right)\ & \ \ &\text{2) Factor the trinomial like you would factor}\ ax^2+bx+c.\end{align}\)
Notice that in the first example, we could further factor the factor \(x^2-1\) into \(\left(x+1\right)\left(x-1\right)\) using the difference of squares factoring pattern. In the second example, a GCF of \(x\) was factored out first.
Speaking of difference of squares, we can use this with fourth degree difference of squares too.
Just like the expression \(x^2-25\) can be factored into \(\left(x+5\right)\left(x-5\right)\), we can factor \(x^4-25\) into \(\left(x^2+5\right)\left(x^2-5\right)\).
Each of the factoring patterns described above could also be used with any higher even degree polynomial as well.
Factor each of the following expressions.
By factoring polynomial expressions, we can see the structure of that expression differently and identify roots and multiplicities of those roots. While we will not cover every possible type of factoring in this section, we will identify some of the most common. In the next target where we solve polynomial equations and find roots of polynomial functions, we will use this factoring to help us.
As we talked about when factoring quadratic expressions (polynomials of degree \(2\)), we first want to always look for a greatest common factor (GCF) and factor that out if there is one. We will incorporate this into many of the examples below.
Fourth Degree Trinomial
If we look at the expression \(x^4-5x^2-6\), hopefully you will notice that it has a similar structure to \(x^2-5x-6\). Just like that expression can be factored into \(\left(x-6\right)\left(x+1\right)\), we can factor \(x^4-5x^2-6\) into \(\left(x^2-6\right)\left(x^2+1\right)\).
Factor the following expressions.
1. \(3x^4+2x^2-5\)
Solution:
\(\begin{align}&\left(3x^2+5\right)\left(x^2-1\right)\ & \ \ &\text{1) Factor the trinomial like you would factor}\ ax^2+bx+c.\\
&\left(3x^2+5\right)\left(x+1\right)\left(x-1\right)\ & \ \ &\text{2) Factor}\ \left(x^2-1\right)\ \text{ with the difference of squares pattern.}\end{align}\)
2. \(x^5-8x^3+12x\)
Solution:
\(\begin{align}&x(x^4-8x^2+12)\ & \ \ &\text{1) Factor out the GCF.}\\
&x\left(x^2-6\right)\left(x^2-2\right)\ & \ \ &\text{2) Factor the trinomial like you would factor}\ ax^2+bx+c.\end{align}\)
Notice that in the first example, we could further factor the factor \(x^2-1\) into \(\left(x+1\right)\left(x-1\right)\) using the difference of squares factoring pattern. In the second example, a GCF of \(x\) was factored out first.
Speaking of difference of squares, we can use this with fourth degree difference of squares too.
Just like the expression \(x^2-25\) can be factored into \(\left(x+5\right)\left(x-5\right)\), we can factor \(x^4-25\) into \(\left(x^2+5\right)\left(x^2-5\right)\).
Each of the factoring patterns described above could also be used with any higher even degree polynomial as well.
Factor each of the following expressions.
- \(2x^6+10x^3-28\)
- \(x^8-81\)
Sum of Squares
Back when you first learned to factor quadratics using difference of squares, you would have left an expression such as \(x^2+36\) as it was since it was a sum. There is no way to factor that expression using only real numbers, but now that you now about imaginary numbers you can.
Let’s multiply the two expressions below.
\(\left(x+6i\right)\left(x-6i\right)\)
\(=x^2+6ix-6ix-36i^2\)
\(=x^2+36\) (recall that \(i^2=-1\))
Anytime we have a sum of squares in the form \(a^2+b^2\), we can factor it using imaginary numbers.
\(a^2+b^2=\left(a+bi\right)\left(a-bi\right)\)
Factor each of the following expressions.
- \(25x^4+49\)
- \(x^6+4\)
Sum and Difference of Cubes
The last type of factoring we will look at is the sum and difference of cubes factoring pattern. This works with expressions in the form of \(a^3+b^3\) or \(a^3-b^3\). Before we get into this, let’s review the perfect cubes. You will want to be very familiar with these numbers so you can identify when this type of factoring is appropriate.
\(1^3=1\), \(2^3=8\), \(3^3=27\), \(4^3=64\), \(5^3=125\), \(6^3=216\), \(7^3=343\), …
The video below will explain how we can get to these factoring patterns and also how to factor these without necessarily memorizing the formulas.
The last type of factoring we will look at is the sum and difference of cubes factoring pattern. This works with expressions in the form of \(a^3+b^3\) or \(a^3-b^3\). Before we get into this, let’s review the perfect cubes. You will want to be very familiar with these numbers so you can identify when this type of factoring is appropriate.
\(1^3=1\), \(2^3=8\), \(3^3=27\), \(4^3=64\), \(5^3=125\), \(6^3=216\), \(7^3=343\), …
The video below will explain how we can get to these factoring patterns and also how to factor these without necessarily memorizing the formulas.
Sum of Cubes: \(a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)
Difference of Cubes: \(a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\)
Difference of Cubes: \(a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)\)
Example 1: Factor \(x^3+64\).
First recognize that \(x^3+64\) is the same as \(x^3+4^3\) which is in the form \(a^3+b^3\) with \(a=x\) and \(b=4\). Using the factoring pattern for the sum of cubes we get \(\left(x+4\right)\left(x^2-x\left(4\right)+4^2\right)\) which simplifies to \(\left(x+4\right)\left(x^2-4x+16\right)\).
So \(x^3+64=\left(x+4\right)\left(x^2-4x+16\right)\).
Example 2: Factor \(8x^3-27\).
Quick Check
Factor each of the following expressions completely.
Quick Check Solutions
Factor each of the following expressions completely.
- \(27x^3+125\)
- \(4x^4-9\)
- \(x^5+10x^3-11x\)
Quick Check Solutions