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Factoring Binomials: How to Break Down Algebraic Expressions

factoring binomials: Two girls writing on a whiteboard

Factoring binomials means breaking down a binomial expression into its simplest form. Binomials are algebraic expressions with two terms. When factoring binomials, you are required to separate the expression into two simpler expressions surrounded by parentheses:

factoring binomials: Mathematical equation

In this article, we’ll cover how to factor binomials and trinomials.

What Are Binomials?

All of the following expressions are examples of binomials:

  • factoring binomials: Example of a binomial

  • factoring binomials: Example of a binomial

  • factoring binomials: Example of a binomial

  • factoring binomials: Example of a binomial

As you can see, any algebraic expression with two terms is a binomial. Binomials can include whole numbers, negative numbers, decimals, or exponents of any value. And it doesn't matter whether the first term or the second term has a variable or not. The only real rule of a binomial is that it has two terms and at least one of them has a variable such as x.

A Guide to Factoring Binomials

Here is an example of a factorable binomial:

factoring binomials: Example of a factorable binomial

The algebraic expression above is an example of a binomial that can be factored, or put in its simplest form because you can take the square root of both x² and 9.

To factor binomials with exponents to the second power, take the square root of the first term and of the coefficient that follows. We’ll look at each part of the binomial separately.

factoring binomials: Mathematical equation

factoring binomials: Mathematical equation

In this binomial, you're subtracting 9 from x². Factoring a binomial that uses subtraction to split up the square root of a number is called the difference of two squares. Since multiplying a negative by a positive equals a negative number, the factorization of this binomial will have to include a positive and a negative 3. When you simplify this binomial, you get this factored form:

factoring binomials: Factored form of a binomial

Binomials That Can't Be Factored

Though many different types of expressions can be classified as binomials, not all of them can be factored. In order to be factorable, a binomial has to have a difference of two squares, a difference of cubes, a sum of cubes, or a greatest common factor. We’ll explain the latter three terms below.

The following is an example of a non-factorable binomial:

factoring binomials: Example of a non-factorable binomial

Here is why this binomial expression cannot be factored:

  1. Unfactorable leading coefficient: The leading coefficient, which is the number written in front of the variable with the largest exponent, is 3. Since 3 is a prime number whose square or cubed root cannot be taken, you can't break this binomial up into two expressions.

  2. There is no greatest common factor (GCF): A GCF is a factor that both terms within the binomial expression have in common. Since there is no common factor between 3 and 14, they can't be divided into two expressions.

Difference and Sum of Cubes

When a binomial doesn’t have a GCF or difference of two squares, it has to have either a difference of cubes or a sum of cubes in order to be factorable. These types of cubed binomial expressions must be written in the following format:

factoring binomials: Formula for a difference of cubes

factoring binomials: Formula for a sum of cubes

In this expression, a and b represent coefficients. Like any other binomial, this cubed expression has to include at least one variable, such as x, to be factorable. The factored form will be separated into a two-term expression and a three-term expression:

factoring binomials: Factored form of the difference of cubes

Factored form of the difference of cubes

Factored form of the sum of cubes

Factored form of the sum of cubes

Let’s use these formulas to factor this difference of cubes expression:

Mathematical expression

Let’s check if coefficients 27 and 64 can be cubed. If so, a will equal the cubed root of 27, and b will equal the cubed root of 64:

Mathematical expression

Mathematical expression

Since both of these numbers can be cubed, let’s plug in the values of a and b to the factored form for the difference of cubes:

Mathematical expression

Factoring Trinomials

Binomials are part of a larger group of expressions called polynomials. Other examples of polynomials are monomials, an expression with only one term, and trinomials, an expression with three terms. Here are examples of each:

Monomial: Example of a monomial

Trinomial: Example of a trinomial

In order to factor trinomials, you have to find a common factor between all three terms. By splitting up the middle term of the trinomial above into 2x and 5x, you can find the greatest common denominator of the first term x^2+2x and the last term 5x+10, respectively:

Example of a trinomial

Mathematical expression

Mathematical expression

Take the outside expression of x+5 and the inner expression of x+2 to get the factored form of this trinomial:

Factored form of the trinomial

The Art of Factoring Binomials

By simplifying a two-term expression, you're putting your knowledge of greatest common denominators and square roots to work. And though factoring binomials takes practice, mastering it makes it easier to factor more complex polynomial expressions.

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