3-formula-solution.jpeg "(a+b)3 Formula Solution")
(a+b)^3 Formula Solution
The (a+b)^3 formula is a fundamental concept in algebra, and it is used to expand the cube of a binomial expression. In this article, we will explore the (a+b)^3 formula, its derivation, and some examples to illustrate its application.
Derivation of the (a+b)^3 Formula
The (a+b)^3 formula can be derived using the binomial theorem, which states that:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + ... + b^n
where n is a positive integer.
To derive the (a+b)^3 formula, we can start with the binomial theorem and set n = 3:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Simplifying the expression, we get:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
which is the (a+b)^3 formula.
** Formula:**
The (a+b)^3 formula is:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Examples:
Example 1:
Expand (x+2)^3 using the (a+b)^3 formula.
(x+2)^3 = x^3 + 3x^2(2) + 3x(2)^2 + 2^3
= x^3 + 6x^2 + 12x + 8
Example 2:
Expand (a-3)^3 using the (a+b)^3 formula.
(a-3)^3 = a^3 - 3a^2(3) + 3a(3)^2 - 3^3
= a^3 - 9a^2 + 27a - 27
Example 3:
Expand (2x+1)^3 using the (a+b)^3 formula.
(2x+1)^3 = (2x)^3 + 3(2x)^2(1) + 3(2x)(1)^2 + 1^3
= 8x^3 + 12x^2 + 6x + 1
Conclusion:
In this article, we have derived and applied the (a+b)^3 formula to expand the cube of a binomial expression. The formula is a powerful tool for simplifying complex algebraic expressions and has numerous applications in mathematics, physics, and engineering.
