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Cube of a Binomial
A binomial is an algebraic expression that has two terms in its simplified form. The word 'cube' of a number refers to a base raised to the power of 3. In this article, we will be studying the cube of a binomial which means a binomial being multiplied by itself 3 times. We will further be learning about the identities and formulas associated with the cube of a binomial.
1.  What is the Cube of a Binomial? 
2.  Cube of a Binomial Formula 
3.  How to Solve Cube of a Binomial? 
4.  FAQs on Cube of a Binomial 
What is the Cube of a Binomial?
The cube of a binomial is defined as the multiplication of a binomial 3 times to itself. We know that cube of any number 'y' is expressed as y × y × y or y^{3}, known as a cube number. Therefore, given a binomial which is an algebraic expression consisting of 2 terms i.e., a + b, the cube of this binomial can be either expressed as (a + b) × (a + b) × (a + b) or (a + b)^{3}.
Cube of a Binomial Formula
We will now be looking into the cube of a binomial formula. There are two formulas of the cube of a binomial depending on the sign between the terms. Those are given below.
In the case of the cube of a binomial with an addition sign between the terms, we use the first formula which can be derived by multiplying the terms.
(a + b)^{3} = (a + b) (a + b) (a + b)
= (a^{2} + 2ab + b^{2}) (a + b)
= a^{3} + 3a^{2}b + 3ab^{2} + b^{3}
= a^{3} + 3ab(a + b) + b^{3}
Thus, the cube of the sum of a binomial can be expressed as: (a + b)^{3} = a^{3} + 3ab(a + b) + b^{3}.
When it comes to the cube of a binomial with a subtraction sign in between, i.e a  b, we use the second formula  (a  b)^{3} = a^{3}  3ab(a  b)  b^{3}.
(a  b)^{3} = (a  b) (a  b) (a  b)
= (a^{2}  2ab + b^{2}) (a  b)
= a^{3}  3a^{2}b + 3ab^{2}  b^{3}
= a^{3}  3ab(a  b)  b^{3}
Thus, the cube of a binomial with a subtraction sign between the terms can be expressed as:^{ }(a  b)^{3} = a^{3}  3ab(a  b)  b^{3}.
How to Solve Cube of a Binomial?
Let's see the steps to solve the cube of the binomial (x + y).
Step 1: First write the cube of the binomial in the form of multiplication (x + y)^{3} = (x + y)(x + y)(x + y).
Step 2: Multiply the first two binomials and keep the third one as it is.
(x + y)^{3} = (x + y)(x + y)(x + y)
(x + y)^{3} = [x(x + y) + y(x + y)](x + y)
(x + y)^{3} = [x^{2 }+ xy + xy +y^{2}](x + y)
(x + y)^{3} = [x^{2 }+ 2xy + y^{2}](x + y)
Step 3: Multiply the remaining binomial to the trinomial so obtained
(x + y)^{3} = [x^{2 }+ 2xy + y^{2}](x + y)
(x + y)^{3} = x(x^{2 }+ 2xy + y^{2}) + y(x^{2 }+ 2xy + y^{2})
(x + y)^{3} = x^{3} + 2x^{2}y + xy^{2} + x^{2}y + 2xy^{2} + y^{3}
(x + y)^{3} = x^{3} + 3x^{2}y + 3xy^{2} + y^{3}
(x + y)^{3} = x^{3} + y^{3 }+ 3x^{2}y + 3xy^{2}
(x + y)^{3} = x^{3} + y^{3} + 3xy(x + y)
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Check these articles related to the concept of the cube of a binomial.
Cube of a Binomial Examples

Example 1: Find the cube of the binomial (3x + 2y).
Solution: We know that for a given binomial (a + b), the cube of the binomial (a + b)^{3} = a^{3} + 3ab(a + b) + b^{3}. We will now be using this formula to evaluate (3x + 2y)^{3}.
Replacing a = 3x and b = 2y in the above formula we get,
(3x + 2y)^{3} = (3x)^{3} + (2y)^{3} + 3(3x)(2y)(3x + 2y)
= 27x^{3} + 8y^{3} + 18xy(3x + 2y)
= 27x^{3} + 8y^{3} + 54x^{2}y + 36xy^{2}
Thus, the cube of the binomial (3x + 2y) is 27x^{3} + 8y^{3} + 54x^{2}y + 36xy^{2}.

Example 2: If the value of (p + q) = 6 and pq = 8, find the value of p^{3} + q^{3}.
Solution: We know that, according to the cube of a binomial formula,
Sum of cubes, a^{3} + b^{3} = (a + b)^{3}  3ab(a + b)
Replacing a = p and b = q, we get,
p^{3} + q^{3} = (p + q)^{3 } 3pq(p + q)
Given that, (p + q) = 6 and pq = 8.
Substituting these in the above formula we get,
p^{3} + q^{3 }= 6^{3}  3 × 8 × 6
= 216  144
= 72
Thus, the value of p^{3} + q^{3} is 72.
FAQs on Cube of a Binomial
What is Cube of a Binomial?
A cube of a binomial is multiplying the binomial three times to itself. For example: (y + z)^{3} = (y + z) × (y + z) × (y + z).
How to Expand Cube of a Binomial?
Cube of a binomial can be expanded using the identities:
(a + b)^{3} = a^{3} + 3ab(a + b) + b^{3}
(a  b)^{3} = a^{3}  3ab(a  b)  b^{3}
What is the Product of the Cube of a Binomial?
The product of the cube of a binomial is defined as multiplying the binomial 3 times with itself and expanding them to find the product as shown: (p + q)^{3} = (p + q) × (p + q) × (p + q) = p^{3} + 3p^{2}q + 3pq^{2} + q^{3}.
What is the general form of the Cube of a Binomial?
The general form of the cube of a binomial is given as: (x + y)^{3} = (x + y)(x + y)(x + y) = x^{3} + 3x^{2}y + 3xy^{2 }+ y^{3}.
What are the Steps in Solving Cube of a Binomial?
The steps to solve a cube of a binomial are given below:
Step 1: First write the cube of the binomial in the form of multiplication (p + q)^{3} = (p + q) × (p + q) × (p + q).
Step 2: Multiply the first two binomials and keep the third one as it is.
Step 3: Multiply the remaining binomial to the trinomial so obtained.
How do you Find the Cube of a Binomial?
A cube of a binomial can be found by multiplying to itself three times. Or we can find the cube by using identities given below:
(a + b)^{3} = a^{3} + 3ab(a + b) + b^{3}
(a  b)^{3} = a^{3}  3ab(a  b)  b^{3}
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