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a plus b cube formula
One of the Algebraic Identities, a plus b cube formula is used to find the cube of a binomial. This algebraic expression is also used to factorize some special types of trinomials.
In this article, the learners will get to know the formula (a+b)3 and comprehend it with supportive examples at the end.
A plus B cube formula
(a+b)3 = a3 + 3a2b + 3ab2 + b3
Derivation of the (a+b)3
To find the cube of a binomial,
we will just multiply (a + b)(a + b)(a + b). (a + b)3 formula is also an identity.
It holds true for every value of a and b.
The (a + b)3 is given as,
(a + b)3 = (a + b)(a + b)(a + b)
= (a2 + 2ab + b2)(a + b)
= a3 + a2b + 2a2b + 2ab2 + ab2 + b³
= a3 + 3a2b + 3ab2 + b³
= a3 + 3ab(a+b) + b3
Therefore, (a + b)3 formula is:
(a + b)3 = a3 + 3a2b + 3ab2 + b3.
Proof of formula in Algebraic Method
The a plus b whole cube formula can be derived in algebraic approach by multiplying three same sum basis binomials. .
Product form of Binomials
Multiplying the binomial by itself three times is the mathematical meaning of the cube of the binomial . So, the A plus B whole cube can be expressed in product form by multiplying three same binomials.
Multiplying three same binomials is a special case in mathematics. Hence, the product of them is often called the special product of binomials.
Multiplying the Algebraic expressions
If It is not possible to multiply all three same binomials at a time, then another method is to
multiply any two binomials first and then multiply the remaining two factors for getting expansion of identity in algebraic approach.
Simplified form of the Expansion
The expansion of the cube of the sum of terms can also be written as the following simplified form.
(a + b)3 = a3 + 3a2b + 3ab2 + b3.
(a + b)3 – Solved Examples using the given formula :
Ques 1: Solve the following expression using suitable algebraic identity:
(2x + 3y)3
Solution:
To find: (2x + 3y)3
Using (a + b)3 Formula,
(a + b)3 = a3 + 3a2b + 3ab2 + b3
= (2x)3 + 3 × (2x)2 × 3y + 3 × (2x) × (3y)2 + (3y)3
= 8x3 + 36x2y + 54xy2 + 27y3 (answer).
Example 2: Find the value of x3+ 8y3
if x + 2y = 6 and xy = 2.
Solution:
To find: x3 + 8y3
Given: x + 2y = 6 and xy = 2
Using (a + b)3 formula,
Here, a = x; b = 2y
Therefore,
(x + 2y)3 = x3+ 3 × x2 × (2y) + 3 × x × (2y)2 + (2y)3
(x + 2y)3 = x3 + 6x2y + 12xy2 + 8y3
63 = x3+ 6xy(x + 2y) + 8y3 ( putting the value of (x+2y)
216 = x3 + 6 × 2 × 6 + 8y3
x3 + 8y3 = 144
