A Cube Plus B Cube

The formula of a cube plus b cube is also known as the formula of the sum of cubes of the two numbers. To find the sum of cubes of two numbers, we can directly use this algebraic formula.  A cube plus b cube is a binomial that can be factorised using the formula. Let us see the formula and solved examples in this article.

A cube plus B cube Formula

The formula for “a cube plus b cube” is given below:

a³ + b³ = (a + b) (a² – ab + b²)

It is also said to be a cube plus b cube identity.

Proof of a cube plus b cube formula

By the formula of a plus b whole cube (a + b)³, we know;

(a + b)³ = a³ + b³ + 3ab (a + b)

Subtracting 3ab (a + b) from both sides, we get;

(a + b)³ – 3ab(a + b) = a³ + b³

Taking (a + b) common, we get;

(a + b) [(a + b)² – 3ab] = a³ + b³

We can write the above expression as;

a³ + b³ = (a + b) [a² + b² + 2ab – 3ab]   (By the identity: (a + b)² = a² + b² + 2ab)

a³ + b³ = (a + b) (a² – ab + b²)

This is the formula for a cube plus b cube.

Solved Examples On A Cube Plus B Cube

Example 1: Factor x³ + 8.

Answer: x³ + 8 can be written as x³ + 2³

Now, x³ + 2³ is in the form of a³ + b³.

Using the formula, a³ + b³ = (a + b) (a² – ab + b²), we get;

x³ + 2³ = (x+2) (x² – 2x + 2²)

x³ + 8 = (x+2) (x² – 2x + 4)

Example 2: Factor the expression 27x³ + 64.

Answer: 27x³ + 64 can be written as (3x)³ + 4³

On comparing with a cube plus b cube identity, a³ + b³ = (a + b) (a² – ab + b²), we have;

a = 3x and b = 4

Therefore, (3x)³ + 4³ = (3x + 4) [(3x)² – (3x).(4) + 4²]

27x³ + 64 = (3x + 4) [9x² – 12x + 16]

Example 3: Find the value of x³ + y³ if the value of the sum x + y = 4 and the product xy = 5.

Answer: We can write,

x³ + y³ in terms of (x + y) and xy using the formula.

x³ + y³ = (x + y)³ – 3xy(x + y)

Substitute 4 for (x + y) and 5 for xy.

x³ + y³ = 4³ – 3(5)(4)

= 64 – 60

= 4

Video Lesson on Algebraic Expansion