What Is the (a + b)3 Formula?
In algebra, the (a + b)3 formula allows us to find the cube of a binomial and helps us to simplify expressions.
The formula for the cube of the sum of two terms, (a + b)3, can be expanded using the binomial theorem or by applying the distributive property multiple times. The result is:
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(a + b)3 = a3 + 3a2b + 3 ab2 + b3
This formula shows how to find the cube of the sum of a and b by raising both terms to the power of 3, along with the combinations involving a and b raised to the power of 2 and 1.
(a + b)3 Formula
The (a + b) whole cube formula or (a + b)3 formula represents the cube of a binomial. It represents the result of cubing the sum of ‘a’ and ‘b’.
(a + b)3 = a3 + 3a2b + 3ab2 + b3
It’s important to note that this formula holds true for any values assigned to both ‘a’ and ‘b.’
Derivation of (a+b)3 Formula
To understand the derivation of the a + b whole cube formula, let’s start by expanding the expression (a + b)3.
We can achieve this by multiplying the binomial (a + b) by itself three times:
(a + b)3 = (a + b)(a + b)(a + b)
On expanding this expression, we get
(a + b)3 = (a2 + 2ab + b2)(a + b)
Let’s use the distributive property to expand the above expression.
(a + b)3 = a3 + a2b + 2a2b + 2ab2 + ab2 + b3
On combining like terms, we arrive at the final form of the (a + b)3 formula:
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Alternatively, we can express the (a + b)3 formula as:
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(a + b)3 = a3 + 3ab (a + b) + b3
This derivation showcases the algebraic manipulation required to arrive at the (a + b)3 formula.
How to Use the (a+b)3 Formula
Let’s use the formula (a + b)3 = a3 + 3a2b + 3ab2 + b3 to expand some expressions.
Example: (x + 2y)3
Here, a = x, b = 2x
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(x + 2y)3 = x3 + 3x2 (2y) + 3x(2y)2 + (2y)3
(x + 2y)3 = x3 + 6x2y + 3x(4y2) + 8y3
(x + 2y)3 = x3 + 6x2y +12xy2 + 8y3
Facts about the (a+b)3 Formula
Conclusion
The (a+b)3 formula is a powerful tool in algebra for expanding and simplifying cubic binomials. By understanding its derivation, properties, and applications, you can enhance your problem-solving skills and tackle a wide range of mathematical challenges. Let’s solve some (a + b)3 formula examples and practice problems to understand the concept better.
Solved Examples on the (a+b)3 Formula
Example 1. Expand (x + 2)3.
Solution:
Using the (a + b)3 formula, we have:
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(x + 2)3 = x3 + 3x2(2) + 3x(2)2 + 23
Simplifying each term, we get
(x + 2)3 = x3 + 6x2 + 12x + 8
Example 2: Evaluate 143 using the (a + b)3 formula.
Solution:
Hare, we can write
143 = (10 + 4)3
Here, a = 10, b = 4
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(10 + 4)3 = 103 + 3(10)2(4) + 3(10)(4)2 + (4)3
(10 + 4)3 = 1000 + 3(100)(4) + 3(10)(16) + 64
(10 + 4)3 = 1000 + 1200 + 480 + 64
(10 + 4)3 = 2744
143 = 2744
Example 3: Find the coefficient of the term a²b in the expansion of (3a + 2b)³.
Solution:
Using the (a + b)3 formula, we get
(3a + 2b)3 = (3a)3 + 3(3a)2(2b) + 3(3a)(2b)2 + (2b)3
(3a + 2b)3 = 27a3 + 3(9a2)2b + 3(3a)(4b2) + 8b3
(3a + 2b)3 = 27a3 + 54a2b +36ab2 + 8b3
So, the coefficient of the term involving “a²b” is 54.
Practice Problems on the (a + b)3 Formula
Frequently Asked Questions about the (a + b)3 Formula
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