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 $a + b$ .

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

\left( a + b \right)^3 =

= \left( a + b \right) \cdot \left( a + b \right) \cdot \left( a + b \right)

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:

= \left( a^2 + 2ab + b^2 \right) \cdot \left( a + b \right)

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

= a^3 + a^2b + 2a^2b + 2ab^2 + ab^2 + b^3

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

= a^3 + 3a^2b + 3ab^2 + b^3

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

Definition

Cube of a Binomial

The cube of a binomial $(A + B)^3$ , also known as the cube of a binomial with like signs, is given by the polynomial:

\left( A + B \right)^3 = A^3 + 3A^2B + 3AB^2 + B^3

Cube of a Binomial with Unlike Signs

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

\left( a - b \right)^3 =

= \left( a - b \right) \cdot \left( a - b \right) \cdot \left( a - b \right)

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

= \left( a^2 - 2ab + b^2 \right) \cdot \left( a - b \right)

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

= a^3 - a^2b - 2a^2b + 2ab^2 + ab^2 - b^3

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

= a^3 - 3a^2b + 3ab^2 - b^3

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

Definition

Cubing a Binomial with Unlike Signs

The cube of a binomial with unlike signs $(A - B)^3$ is the following polynomial:

\left( A - B \right)^3 = A^3 - 3A^2B + 3AB^2 - B^3

In the result, the two terms where $B$ has an odd exponent are negative.

Examples

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

Example

Example 1

Calculate the cube of $x + 2$ .

First, let's try to compute the result directly without using the formula. So, we multiply the binomial by itself twice:

\left( x + 2 \right)^3 =

= \left( x + 2 \right) \cdot \left( x + 2 \right) \cdot \left( x + 2 \right)

= \left( x^2 + 4x + 4 \right) \cdot \left( x + 2 \right)

= x^3 + 2x^2 + 4x^2 + 8x + 4x + 8

= x^3 + 6x^2 + 12x + 8

Now, let's apply the formula for the cube of a binomial directly:

\left( x + 2 \right)^3 =

= x^3 + 3x^2(2) + 3x(2^2) + 2^3

= x^3 + 6x^2 + 12x + 8

The result matches the one we obtained earlier.

Example

Example 2

Calculate the cube of $2a^2 - 3b$ .

First, let's try to compute the result directly without using the formula. So, we multiply the binomial by itself twice:

\left( 2a^2 - 3b \right)^3 =

= \left( 2a^2 - 3b \right) \cdot \left( 2a^2 - 3b \right) \cdot \left( 2a^2 - 3b \right)

= \left( 4a^4 - 12a^2b + 9b^2 \right) \cdot \left( 2a^2 - 3b \right)

= 8a^6 - 12a^4b - 24a^4b + 36a^2b^2 + 18a^2b^2 - 27b^3

= 8a^6 - 36a^4b + 54a^2b^2 - 27b^3

Now, let's apply the formula for the cube of a binomial with unlike signs directly:

\left( 2a^2 - 3b \right)^3 =

= (2a^2)^3 - 3(2a^2)^2(3b) + 3(2a^2)(3b)^2 - (3b)^3

= 8a^6 - 36a^4b + 54a^2b^2 - 27b^3

The result matches the one we obtained earlier.

In Summary

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

\left( A + B \right)^3 = A^3 + 3A^2B + 3AB^2 + B^3

Since we are dealing with powers of $A$ and $B$ 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 $B$ has an odd exponent have a negative sign:

\left( A - B \right)^3 = A^3 - 3A^2B + 3AB^2 - B^3