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Home >> Standard Identities & their applications >> (a + b)^{3} = a^{3} + b^{3} + 3ab(a + b) >> (A + B) Cube
Before you understand (a + b)^{3} = a^{3} + b^{3} + 3ab(a + b), you are advised to read:
How to multiply Variables ?
How to multiply Constant and Variable ?
Multiplication of Polynomials ?
What is Exponential Form and Laws of Exponents ?
How identity of (a + b)^{3} = a^{3} + b^{3} + 3ab(a + b) is obtained
Taking LHS of the identity:
(a + b)^{3}
This can also be written as:
= (a + b) (a + b) (a + b)
Multiply first two binomials as shown below:
= { a(a + b) + b(a + b) } (a + b)
= { a^{2} + ab + ab + b^{2} } (a + b)
Rearrange the terms in curly braces and we get:
= { a^{2} + b^{2} + ab + ab } (a + b)
= { a^{2} + b^{2} + 2ab } (a + b)
Multiply trinomial with binomial as shown below:
= a^{2}(a + b) + b^{2}(a + b) + 2ab(a + b)
= a^{3} + a^{2}b + ab^{2} + b^{3} + 2ab(a + b)
= a^{3} + b^{3} + a^{2}b + ab^{2} + 2ab(a + b)
= a^{3} + b^{3} + a^{2}b + ab^{2} + 2ab(a + b)
Take ab common from the above red highlighted terms and we get:
= a^{3} + b^{3} + ab(a + b) + 2ab(a + b)
Add like terms and we get:
= a^{3} + b^{3} + 3ab(a + b)
Or we can further solve this to get:
a^{3} + b^{3} + 3a^{2}b + 3ab^{2}
Hence, in this way we obtain the identity i.e. (a + b)^{3} = a^{3} + b^{3} + 3ab(a + b) = a^{3} + b^{3} + 3a^{2}b + 3ab^{2}
Let's try some examples on this identity
Example 1: Solve (2a + 3b)^{3}
Solution: This proceeds as:
Given polynomial (2a + 3b)^{3} represents identity i.e. (a + b)^{3}
Where a = 2a and b = 3b
Now apply values of a and b on the identity i.e. (a + b)^{3} = a^{3} + b^{3} + 3ab(a + b) and we get:
(2a + 3b)^{3} = (2a)^{3} + (3b)^{3} + 3(2a) (3b)(2a + 3b)
Expand the exponential forms and we get:
= 8a^{3} + 27b^{3} + 3(2a) (3b)(2a + 3b)
Solve multiplication process and we get:
= 8a^{3} + 27b^{3} + 18ab(2a + 3b)
Hence, (2a + 3b)^{3} = 8a^{3} + 27b^{3} + 18ab(2a + 3b)
Example 2: Solve (5x + 6y)^{3}
Solution: This proceeds as:
Given polynomial (5x + 6y)^{3} represents identity i.e. (a + b)^{3}
Where a = 5x and b = 6y
Now apply values of a and b on the identity i.e. (a + b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2} and we get:
(5x + 6y)^{3} = (5x)^{3} + (6y)^{3} + 3(5x)^{2} (6y) + 3(5x) (6y)^{2}
Expand the exponential forms and we get:
= 125x^{3} + 216y^{3} + 3(25x^{2})(6y) + 3(5x) (36y^{2})
Solve multiplication process and we get:
= 8a^{3} + 27b^{3} + 450x^{2}y + 540xy^{2}
Hence, (5x + 6y)^{3} = 8a^{3} + 27b^{3} + 450x^{2}y + 540xy^{2}


