Introduction
Factoring is an important skill for math students because it helps with solving equations, simplifying expressions, and understanding how functions behave. It is one of the basic building blocks of algebra.
When we factor a polynomial, our goal is to change a long expression made of added or subtracted terms into a product of smaller pieces. This is similar to breaking the number 12 into 2⋅2⋅3. These smaller numbers multiply together to make 12, and they help us understand the structure of the number better.
We want to do the same thing with polynomials. Breaking a polynomial into smaller factors makes it easier to work with and easier to solve.
When you begin factoring a polynomial, the first step is to check for a Greatest Common Factor (GCF). The GCF is a single term (a monomial) that divides evenly into every term of the polynomial. We find the GCF by breaking each term into its prime factors and looking for what they all share. Once the GCF is found, we factor it out before moving on to any other method.
Check out the guide below, and join us to learn more and ask questions about this topic in the MAT-126: Modern Problem Solving workshop. Click here to see the full group session schedule.
Example
Given:
4x2y3 − 6xy2 + 2x3y4
We can determine that the GCF they all have in common is 2xy2 We will divide each term by this monomial.
2xy − 3y + xy2
When you divide each term by the GCF (also known as factoring out the GCF), you will get 2xy – 3 + xy2
When writing the result, the GCF is placed in front of a set of parentheses and the rest is left inside the parentheses as follows:
2xy2(2xy - 3y + xy2)
To confirm you did it correctly, verify:
- The polynomial inside the parentheses no longer has something in common. This illustrates you chose the GREATEST common factor
- Re-distribute your GCF through the parentheses to verify you can get the original question back again, confirming you didn’t make any computational errors
Practice Problems:
- 2x2 + 8x
- 8x–12
- 5x3y+x2
- 12x2–9x+21
- –4a2b3–6a4b2–18a3b5
- 7x2–30
For more guides on math topics, visit our YouTube Channel “All Math Videos” playlist.
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