Cube of Binomial

Cube of Binomial as the name suggests is the third power of any binomial expression. Cube of Binomial follows a specific formula, which is (a + b)³ = a³ + 3a²b + 3ab² + b³) and (a - b)³ = a³ - 3a²b + 3ab² - b³), where (a) and (b) are the terms of the binomial.

In this article, we will learn about the sum of cubes formula, the difference of cubes formula, and how to find a cube of binomial. At the end of this article, we have provided solved numerical questions for better understanding.

👁 Cube-of-Binomial

What is Cube of Binomial?

Cube of a binomial refers to the result obtained by raising a binomial expression to the power of 3. This process involves multiplying the binomial by itself twice and expanding the expression, resulting in a trinomial. The general form of the cube of a binomial, (a + b)³, is expressed as a³ + 3a²b + 3ab² + b³, showcasing the coefficients derived from the expansion. Understanding the cube of a binomial is fundamental in algebraic expressions and polynomial manipulations.

Meaning of Cube of Binomial

Cube of a binomial refers to raising a binomial expression to the power of 3.

This process involves multiplying the binomial by itself twice and simplifying the resulting expression.

Formula of Cube of Binomial

The formula for the cube of a binomial a + b and a - b is given by:

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3ab² - b³

Derivation of (a+b)³

(a+b)³ = (a+b)(a+b)(a+b)

⇒ (a²+2ab+b²) (a+b)

⇒ a(a²+2ab+b²) + b(a²+2ab+b²)

⇒a³+2a²b+ab²+a²b+2ab²+b³

⇒ a³+3a²b+3ab²+b³

∴ (a+b)³ = a³+3a²b+3ab²+b³

Derivation of (a-b)³

(a - b)³ = (a - b)(a - b)(a - b)

⇒ (a - b)³ = (a - b)(a - b)(a - b)

⇒ (a - b)³ = (a² - 2ab + b²)(a - b)

Using the distributive property multiply (a² - 2ab + b²) by (a - b):

⇒ (a - b)³ = (a² - 2ab + b²)(a - b) = a(a² - 2ab + b²) - b(a² - 2ab + b²)

Next, distribute (a) and (-b) into each term:

⇒ (a - b)³ = a³ - 2a²b + ab² - a²b + 2ab² - b³

⇒ (a - b)³ = a³ - 3a²b + 3ab² - b³

∴ (a - b)³ = (a³ - 3a²b + 3ab² - b³).

Sum of Cubes Formula

The sum of cubes formula is a special case of the polynomial expansion known as the sum of cubes identity. It states that the sum of two cubes, a³+b³, can be factored into the product of a binomial and a trinomial.

a³ + b³ = (a + b)(a² - ab + b²)

Derivation of Sum of Cubes Formula

To derive a³+b³ using the sum of cubes formula, we start with the formula:

(a+b)³ = a³+ 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b)

⇒ (a+b)³ - 3ab(a + b) = a³ + b³

⇒ [(a+b)² - 3ab](a + b) = a³ + b³

⇒ [a² + b² + 2ab - 3ab](a + b) = a³ + b³

⇒ [a² + b² - ab](a + b) = a³ + b³

Difference of Cubes Formula

The difference of cubes formula states that the difference of two cubes, ( a³ - b³ ), can be factored into (a - b)(a² + ab + b²). This formula is derived by expanding (a - b)(a² + ab + b²) using the distributive property, which results in (a³ - b³). It's a helpful in algebra for factoring expressions involving the difference of two cube terms.

Derivation of Difference of Cubes

To derive a³- b³ using the sum of cubes formula, we start with the formula:

(a - b)³ = a³ - 3a²b + 3ab² - b³ = a³ - b³ - 3ab(a - b)

⇒ (a - b)³ + 3ab(a - b) = a³ - b³

⇒ [(a - b)² + 3ab](a - b) = a³ - b³

⇒ [a² + b² - 2ab + 3ab](a - b) = a³ - b³

⇒ [a² + b² + ab](a - b) = a³ - b³

How to Solve Cube of Binomial?

To calculate cube of binomial, we can use the following steps:

Step 1: Identify the Binomial.

Suppose we have the binomial (a + b).

Step 2: Cube the Binomial.

Use the formula (a + b)³ = a³ + 3a²b + 3ab² + b³ to expand the cube of the binomial.

Step 3: Apply the Binomial Cube Formula.

Substitute the values of ( a ) and ( b ) into the expanded expression.

Step 4: Simplify.

Combine like terms and simplify the expression.

Let's consider an example for the same.

For example, if we have ( a = 2 ) and ( b = 3 ), then:

(2 + 3)³ = 2³ + 3 · 2² · 3 + 3 · 2 · 3² + 3³

⇒ (2 + 3)³ = 8 + 3 · 4 · 3 + 3 · 2 · 9 + 27

⇒ (2 + 3)³ = 8 + 36 + 54 + 27

⇒ (2 + 3)³ = 125

∴ (2 + 3)³ = 125

Read More,

• Squaring a Trinomial
• Types of Polynomials
• Binomial Distribution in Probability

Solved Examples of Cube of Binomial

Example 1: Find the cube of the binomial (x + 2).

Solution:

To find the cube of the binomial (x + 2), we'll apply the formula for the cube of a binomial, which is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here, a = x and b = 2.

Plugging these values into the formula, we get:

(x + 2)³ = x³ + 3x²(2) + 3x(2)² + 2³

= x³ + 6x² + 12x + 8

So, the cube of the binomial (x + 2) is x³ + 6x² + 12x + 8

Example 2: Calculate the cube of the binomial (3y - 4).

Solution:

To calculate the cube of the binomial (3y - 4), we'll use the formula for the cube of a binomial:

(a - b)³ = a³ - 3a²b + 3ab² - b³

Here, a = 3y and b = 4

Plugging these values into the formula, we get:

(3y - 4)³ = (3y)³ - 3(3y)²(4) + 3(3y)(4)² - 4³

= 27y³ - 3(9y²)(4) + 3(3y)(16) - 64

= 27y³ - 108y² + 144y - 64

∴ the cube of the binomial (3y - 4) is 27y³ - 108y² + 144y - 64

Example 3: Determine the value of (2a - 1)³.

Solution:

Using the formula for the cube of a binomial:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here, a = 2a and b = -1. Plugging these values into the formula, we get:

(2a - 1)³ = (2a)³ + 3(2a)²(-1) + 3(2a)(-1)² + (-1)³

= 8a³ - 12a² + 6a - 1

So, the value of (2a - 1)³ is 8a³ - 12a² + 6a - 1

Example 4: Find the cube of the binomial (b + 5).

Solution:

To find the cube of the binomial (b + 5), we'll apply the formula for the cube of a binomial, which is:

(a + b)³ = a³ + 3a²b + 3ab² + b³

Here, a = b and b = 5

Plugging these values into the formula, we get:

(b + 5)³ = b³ + 3b²(5) + 3b(5)² + 5³

= b³ + 15b² + 75b + 125

So, the cube of the binomial (b + 5) is b³ + 15b² + 75b + 125

Practice Questions of Cube of Binomial

Q1. Evaluate (4z - 6)³.

Q2. Find the cube of (m + 7).

Q3. Calculate (2t - 9)³.

Q4. Determine the cube of the binomial (n - 2).

Q5. Find the value of (6p + 1)³