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 y3, 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)
= (a2 + 2ab + b2) (a + b)
= a3 + 3a2b + 3ab2 + b3
= a3 + 3ab(a + b) + b3
Thus, the cube of the sum of a binomial can be expressed as: (a + b)3 = a3 + 3ab(a + b) + b3.
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 = a3 - 3ab(a - b) - b3.
(a - b)3 = (a - b) (a - b) (a - b)
= (a2 - 2ab + b2) (a - b)
= a3 - 3a2b + 3ab2 - b3
= a3 - 3ab(a - b) - b3
Thus, the cube of a binomial with a subtraction sign between the terms can be expressed as: (a - b)3 = a3 - 3ab(a - b) - b3.
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 = [x2 + xy + xy +y2](x + y)
(x + y)3 = [x2 + 2xy + y2](x + y)
Step 3: Multiply the remaining binomial to the trinomial so obtained
(x + y)3 = [x2 + 2xy + y2](x + y)
(x + y)3 = x(x2 + 2xy + y2) + y(x2 + 2xy + y2)
(x + y)3 = x3 + 2x2y + xy2 + x2y + 2xy2 + y3
(x + y)3 = x3 + 3x2y + 3xy2 + y3
(x + y)3 = x3 + y3 + 3x2y + 3xy2
(x + y)3 = x3 + y3 + 3xy(x + y)
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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 = a3 + 3ab(a + b) + b3
(a - b)3 = a3 - 3ab(a - b) - b3
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) = p3 + 3p2q + 3pq2 + q3.
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) = x3 + 3x2y + 3xy2 + y3.
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 = a3 + 3ab(a + b) + b3
(a - b)3 = a3 - 3ab(a - b) - b3