Cube of a Binomial

The cube of a binomial is a notable significant algebraic identity. In this lesson, we will derive the formula to compute the cube of a binomial, both when the signs of the terms are the same and when they are different. We will also look at some examples.

Cube of a Binomial

Let's take a binomial composed of two monomials, for example .

To compute the cube of this binomial, we must multiply the binomial by itself two times, as follows:

Let's try to determine the result. First, we multiply the first binomial by the second. However, this is the square of a binomial, so we can write:

Now, we can expand the product between these two polynomials as:

Next, we can simplify the result by combining like terms:

This result is an important algebraic identity and is called the cube of a binomial.

Cube of a Binomial with Unlike Signs

If the binomial consists of two monomials with unlike signs, for instance , the outcome will differ. Let's try to calculate the cube of this binomial:

Using the square of a binomial again to simplify the product:

As we can observe in this case, the intermediate monomial is negative. Let's expand the product as:

Now we can simplify the result, by adding and subtracting the like terms:

Compared to the previous case, the two terms where appears with an odd exponent are negative. Therefore:

Examples

Now, let's see some examples of cubing a binomial.

In Summary

In this lesson, we studied the notable algebraic identity of the Cube of a Binomial. Given two monomials and , we derived the formula:

Since we are dealing with powers of and with odd exponents, we observed that the result varies if the signs of the two monomials are alike or opposite. Specifically, when the signs are opposite, the two terms where has an odd exponent have a negative sign: