What Is Factoring Binomials?
A binomial is an algebraic expression with two terms. Factoring a binomial means rewriting it as a product. The standard routine is to take out the greatest common factor first, then check for a difference of squares, a sum of cubes or a difference of cubes.
Enter the coefficient and exponent of each term. The calculator removes the common factor, tests each pattern in turn, factors repeatedly when a factor is itself a difference of squares and tells you when the binomial is prime. A reference table lists the binomial factoring formulas for quick revision.
The sections below give the formulas, a variables table, a worked example and answers to common questions about binomial factoring, such as why a sum of squares does not factor.
How to Use the Factoring Binomials Calculator
- Enter the coefficient and exponent of each term (use a negative coefficient for subtraction).
- The calculator removes the GCF.
- It then tests for a difference of squares or a sum or difference of cubes.
- Read the factored form and the steps.
Factoring Binomials Formula
Difference of squares
Sum of cubes
Difference of cubes
Factoring Binomials Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| a, b | The base of each perfect power | for example 3x and 5 in 9x² − 25 |
| GCF | Greatest common factor of the two terms | |
| a² − b² | Difference of squares | (a − b)(a + b) |
| a³ ± b³ | Sum or difference of cubes |
Factoring Binomials Worked Example
Factor 16x⁴ − 81. There is no common factor. Both terms are perfect squares: (4x²)² − 9². So 16x⁴ − 81 = (4x² − 9)(4x² + 9). The factor 4x² − 9 is again a difference of squares, (2x − 3)(2x + 3), so the full factorisation is (2x − 3)(2x + 3)(4x² + 9).
With the example values loaded in the calculator above, the steps are:
- Write the binomial in descending order: 16x⁴ − 81.
- Step 1. The terms share no common factor other than 1.
- Step 2. 16x⁴ − 81 is a difference of squares: (4x²)² − 9², and a² − b² = (a − b)(a + b).
- Step 3. The first factor is itself a difference of squares, so it factors again.
- Answer: 16x⁴ − 81 = (2x − 3)(2x + 3)(4x² + 9).
Result: Factored form = (2x − 3)(2x + 3)(4x² + 9); Pattern used = Difference of two squares.
How to Read a Factored Binomial
Factoring writes the binomial as a product. First take out any common factor, then look for a pattern. 16x⁴ − 81 is a difference of squares, (4x²)² − 9², which factors as (4x² − 9)(4x² + 9).
Check whether any factor can be factored again. 4x² − 9 is another difference of squares, so the full factoring is (2x − 3)(2x + 3)(4x² + 9). A sum of two squares does not factor over the real numbers.
Where Factoring Binomials Is Used
Difference of squares
Factor a² − b² into (a − b)(a + b), including higher powers.
Sums and differences of cubes
Use the cube patterns to factor a³ + b³ and a³ − b³.
Common factors
Take out the greatest common factor first.
Homework
Check each factoring step and the pattern used.
Common Factoring Binomials Mistakes to Avoid
- Always take out the greatest common factor first.
- A sum of squares does not factor over the real numbers. A sum of cubes does.
- After one factoring, check whether a factor can be factored again.
- Multiply your factors back out to check the answer.
Factoring Binomials Terms and Definitions
- Binomial
- An expression with two terms.
- GCF
- The largest factor shared by all terms.
- Prime polynomial
- A polynomial that cannot be factored over the integers.
Factoring Binomials FAQs
How do you factor a binomial?
Take out any common factor, then check for a pattern: difference of squares, sum of cubes or difference of cubes. Factor again if a result allows.
Can a sum of squares be factored?
Not over the real numbers. x² + 9 stays as it is. It factors only if you allow complex numbers.
What is the difference of squares formula?
a² − b² = (a − b)(a + b). For 16x⁴ − 81, a = 4x² and b = 9.
How do I factor a sum of cubes?
Use a³ + b³ = (a + b)(a² − ab + b²). For x³ + 8, a = x and b = 2, which gives (x + 2)(x² − 2x + 4).