Polynomials are algebraic expressions that consist of a combination of variables and coefficients. Factoring polynomials involves breaking down these complex expressions into simpler components, allowing students to work and solve them more easily.
Whether you are a student starting on algebra or calculus or brushing up on GED math skills, knowing how to factor polynomials is a valuable skill that can help you perform well in various exams, including AP Calculus, A-level mathematics, and International Baccalaureate (IB).
Read on to explore some of the most common methods for factoring polynomials that you need to know. If you wonder how to find the inverse of a function, I wrote a detailed article that I encourage you to check out.
What Is a Polynomial?
A polynomial is an expression that involves variables and coefficients and contains one or more terms added, subtracted, and/or multiplied together. For example, 4x2 + 3x – 2 is a polynomial, with 4, 3, and -2 being coefficients. Check out this video to learn more about polynomials.
What Does It Mean to Factor a Polynomial?
Factoring a polynomial means expressing a polynomial as a product of simpler polynomials and/or terms. For example, we can factor the polynomial 6x2 + 11x + 4 as (2x + 1)(3x + 4).
How To Factor Polynomials
Most students, particularly high school students, find factoring polynomials challenging; however, I believe it is an essential skill that every student should master. Here’s how to factor polynomials:
1- Factor Out a Common Term
One of the methods to factor a polynomial is to look for the greatest common factor (GCF) among all the terms. In other words, the GCF refers to the largest factor of two or more expressions, which can be factored out of each term of a polynomial.
To find the GCF, identify the common factors of the coefficients and variables and then choose the one with the highest degree.
For example, in the following polynomials:
- 12x3 + 16x2, the GCF is 4x2. We can then divide each term by the GCF to get 4x2(3x + 4).
- 6x3+12x2, the GCF is 6x2. We can factor this out to get 6x2(x+2).
2- Difference of Squares
The difference of squares method is used when the polynomial contains two perfect squares with a minus sign between them. In other words, if the polynomial is a binomial (i.e., has two terms), we can use the difference of squares or the sum/difference of cube formulas to factor it.
- The difference of squares formula is a2 – b2 = (a + b)(a – b)
- The sum of cubes formula is a3 + b3= (a + b)(a2 – ab + b2)
- And the difference of cubes formula is a3 – b3 = (a – b)(a2 + ab + b2)
For example, we can factor the binomial x2 – 9 as (x + 3)(x – 3) using the difference of squares formula. If you are interested in learning how to factor the difference of squares, I invite you to watch the video below.
3- Factoring Quadratics
Another popular method for factoring polynomials is the quadratic formula. Quadratic expressions are those that have an equation with a squared term and a constant term. The general form of a quadratic equation is ax2 + bx + c = 0, with a, b, and c coefficients. We can use the quadratic formula to factor polynomials that are quadratic in form.
The quadratic formula is x = (-b ± √(b2 – 4ac)) / 2a, where a, b, and c are the coefficients of the polynomial.
For example, we can factor the quadratic polynomial 2x2 + 5x – 3 as (2x – 1)(x + 3/2) using the quadratic formula. However, not all quadratic expressions can be factored in. If the quadratic equation has an imaginary root, it is impossible to find a real solution.
If you want to explore how to solve quadratic equations by factoring, I wrote a whole step-by-step guide that I encourage you to check out.
4- Factoring by Grouping
Another method of factoring polynomials is by grouping. This method works if there are four or more terms in a polynomial and if there are common factors among the terms. To use this method, first group the terms that have common factors and then factor out the GCF of each group.
For example, the polynomial 2x3 + 2x2 – 3x – 3 can be grouped as (2x3 + 2x2) – (3x + 3), and factored as 2x2(x + 1) – 3(x + 1) = (2x2 – 3)(x + 1).
Another example is if we have the polynomial 3x3+6x2+4x+8, we can group the first two terms and the second two terms. We then factor out the GCF of each group, leaving us with the factored form of 3x2(x+2)+4(x+2). Note that the (x+2) can be factored out further.
If you are interested in using factoring polynomials by grouping, I encourage you to check out this video.
5- Completing the Square
The process of completing the square is used to transform a quadratic equation into a perfect square trinomial. This method is useful when solving equations that require the use of the quadratic formula.
The steps involved include taking half the coefficient of the x term, squaring it, and adding it to both sides of the equation.
An example of an equation that can be factored using this method is x2+8x+7, which can be factored as (x+4)2-9. If you want to explore more about the completing the square method, I wrote a whole step-by-step guide that I invite you to read.
What to read next:
- The FOIL Method for Binomial Multiplication: Definition and Examples!
- 17 Maths Websites for High School Students to Get Ahead.
- Understanding the Properties of Quadratic Functions.
- Start by identifying any common factors, and you can either use the following factoring methods:
- Use the difference of squares or sum/difference of cubes formulas for binomials,
- Group the terms for polynomials with four or more terms
- Use the quadratic formula for quadratic polynomials.
I believe that practice makes perfect, meaning that the more you factor polynomials, the easier it will be.