In this article, we are going to introduce a new concept that is Cube of the Sum of Two Binomials terms. We are providing the different problems with a clear explanation on this topic. Follow our page and get full of knowledge on it. Firstly, to find the cube of the sum of two binomials, we need to multiply the binomials term three times. Refer to Solved Examples on Cube of Sum of Two Binomials provided along with Solutions for better understanding of the concept.

## How to find the Cube of Sum of Two Binomials?

For example, (x + y) ^3 = (x + y) (x + y)^2 is the example of a binomial expression.

Here, we have an equation in an algebra like (a + b)^2 = a^2 + 2ab + b^2.

By using the above equation, we can expand the cube term.

(x + y) (x + y)^2 = (x + y) (x^2 + 2xy + y^2).

Multiply the terms (x + y) and (x^2 + 2xy + y^2). Then we get

(x + y) (x^2 + 2xy + y^2) = x (x^2 + 2xy + y^2) + y (x^2 + 2xy + y^2).

= x^3 + 2x^2y + xy^2 + yx^2 + 2xy^2 + y^3.

= x^3 + 3x^2y + 3xy^2 + y^3.

= x^3 +y^3 + 3xy(x + y).

Also, Read: Cube of a Binomial

## Cube of Sum of Two Binomials Examples

**1. Determine the expansion of (x + 2y)^3.**

Solution:

The given expression is (x + 2y)^3.

We have an equation on cubes like (x + y)^3 = x^3 + y^3 + 3xy(x + y).

By comparing the above expression with the (x + y)^3

Here, x = x and y = 2y

Substitute the terms in the equation (x + y)^3

That is, (x + 2y)^3 = x^3 + (2y)^3 + 3x(2y)(x + 2y).

= x^3 + 8y^3 + 6xy(x + 2y).

= x^3 + 8y^3 + 6x^2y + 12xy^2.

Therefore, (x + 2y)^3 is equal to x^3 + 8y^3 + 6x^2y + 12xy^2.

**2. Evaluate (55)^3.**

Solution:

The given one is (55)^3.

We can write it as (50 + 5)^3.

(x + y)^3 = x^3 + y^3 + 3xy(x + y).

By comparing the (50 + 5)^3 with the above expression.

x = 50 and y = 5.

Substitute the values in the expression.

(50 + 5)^3 = (50)^3 + (5)^3 + 3(50)(5)(50 + 5).

= 1,25,000 + 125 + 750(55).

= 1,25,000 + 125 + 41,250.

= 1,66,375.

Therefore, (55)^3 is equal to 1,66,375.

**3. Find the value of 64x^3 + y^3 if 4x + y = 6 and xy = 5.**

Solution:

The given expression is 64x^3 + y^3.

4x + y = 6.

Cube the terms on both sides. Then, we will get

(4x + y)^3 = (6)^3.

We have an equation (x + y)^3 = x^3 + y^3 + 3xy(x + y).

Here, x = 4x and y = y.

Substitute the values in the equation. Then,

(4x)^3 + y^3 + 3(4x)(y)(4x + y) = 216.

64x^3 + y^3 + 12xy(4x + y) = 216.

But 4x + y = 6 and xy = 5.

So, 64x^3 + y^3 + 12(5)(6) = 216.

64x^3 + y^3 + 360 = 216.

64x^3 + y^3 = -144.

Therefore, 64x^3 + y^3 is equal to -144.

**4. If a + 1/a = 3, find the values of a^3 – 1/a^3.**

Solution:

The given term is a + 1/a = 3

Cube the terms on both sides. Then, we will get

(a + 1/a)^3 = (3)^3.

We have an equation (x + y)^3 = x^3 + y^3 + 3xy(x + y).

By comparing the given terms with the equation.

Here, x = a and y = 1/a.

By substituting the terms in the given equation. We will get

(a + 1/a)^3 = a^3 + (1/a)^3 + 3a(1/a)(a + 1/a) = 27.

= a^3 + 1/a^3 +3(a + 1/a) = 27.

We have (a + 1/a) = 3. By substituting this value in the above expression.

a^3 + 1/a^3 + 3(a + 1/a) =27.

a^3 + 1/a^3 + 3(3) = 27.

a^3 + 1/a^3 +9 = 27.

a^3 + 1/a^3 = 27 – 9 = 18.

Therefore, a^3 + 1/a^3 is equal to 18.

**5. Expand the term (2x + y)^3.**

Solution:

The given expression is (2x + y)^3.

we have an equation (x + y)^3 = x^3 + y^3 + 3xy(x + y).

by comparing the (2x + y)^3 with the above equation.

Here, x = 2x and y = y.

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

= 8x^3 + y^3 + 6xy(2x + y).

= 8x^3 + y^3 + 12x^2y + 6xy^2.

The final answer is 8x^3 + y^3 + 12x^2y + 6xy^2.