In algebra, the formula for a³ + b³, also called the sum of cubes, is a useful identity that helps us break down the addition of two cube terms into simpler parts. Instead of directly calculating and adding the cubes of two numbers, we can use a special formula to factor them easily. This formula is written as:

a³ + b³ = (a + b)(a² − ab + b²)

It helps in solving problems faster, especially in algebraic expressions where large numbers or variables are involved. The formula is not just a shortcut, but a helpful tool in simplifying and understanding complex algebraic patterns. In this lesson, we’ll look into how this formula is derived, how it works, and go through a few examples so you can see how to use it clearly and confidently in your math problems.

The Formula for A Cube Plus B Cube

So, what exactly is the formula for “a cube plus b cube”? Let's unveil it:

a 3 + b 3 = (a + b) (a 2 – ab + b 2 )

This is also referred to as the cube plus cube identity.

How is the Formula for a³ + b³ Derived

Let’s begin with the expansion of the cube of a binomial:

(a + b)³ = a³ + b³ + 3ab(a + b)

Now, rearrange the terms:

a³ + b³ = (a + b)³ − 3ab(a + b)

Now take out (a + b) as a common factor:

a³ + b³ = (a + b) × [ (a + b)² − 3ab ]

Now simplify the expression inside the brackets:

(a + b)² = a² + 2ab + b²

Substitute it back:

a³ + b³ = (a + b) × [ a² + 2ab + b² − 3ab ]

Now simplify further:

a³ + b³ = (a + b) × (a² − ab + b²)

Final Formula: a³ + b³ = (a + b)(a² − ab + b²)

Factors of a³ + b³:

The expression a³ + b³ can be factored as:

(a + b)(a² − ab + b²)

This identity helps in factoring the sum of two cubes easily.

The Proof of A Cube Plus B Cube Formula

If you're wondering how this formula is derived, let's walk through its proof using the formula of a plus b whole cube (a + b) 3 :

(a + b) 3 = a 3 + b 3 + 3ab (a + b)

By subtracting 3ab (a + b) from both sides, we get:

(a + b) 3 – 3ab(a + b) = a 3 + b 3

Taking (a + b) common, we get:

(a + b) [(a + b) 2 – 3ab] = a 3 + b 3

Using the identity: (a + b) 2 = a 2 + b 2 + 2ab, we can rewrite the above expression as:

a 3 + b 3 = (a + b) (a 2 – ab + b 2 )

And there you have it, the proof of the formula for a cube plus b cube!

a³ + b³ vs a³ - b³

Examples of A Cube Plus B Cube

Example 1: Factorise the expression y³ + 27.
Solution:
y³ + 27 can be written as y³ + 3³.
Now, y³ + 3³ is in the form of a³ + b³.
Using the identity:
a³ + b³ = (a + b)(a² – ab + b²)
So,
y³ + 3³ = (y + 3)(y² – 3y + 3²)
y³ + 27 = (y + 3)(y² – 3y + 9)

Example 2: Factorise the expression 8z³ + 125.
Solution:
8z³ + 125 can be written as (2z)³ + 5³.
This is in the form a³ + b³.
Using the identity:
a³ + b³ = (a + b)(a² – ab + b²)
Here, a = 2z and b = 5
So,
(2z)³ + 5³ = (2z + 5)[(2z)² – (2z)(5) + 5²]
8z³ + 125 = (2z + 5)(4z² – 10z + 25)

Example 3: Find the value of p³ + q³ if p + q = 3 and pq = 2.
Solution:
Use the identity:
p³ + q³ = (p + q)³ – 3pq(p + q)
Substitute values:
p + q = 3 and pq = 2
p³ + q³ = 3³ – 3(2)(3)
p³ + q³ = 27 – 18
p³ + q³ = 9