%5E3-formula-derivation.jpeg "(a+b)^3 Formula Derivation")
The Derivation of (a+b)^3 Formula
The expansion of (a+b)^3 is a fundamental concept in algebra, and it's essential to understand the derivation of this formula to apply it in various mathematical problems. In this article, we'll explore the step-by-step derivation of the (a+b)^3 formula.
The Binomial Theorem
The binomial theorem is a mathematical formula that describes the expansion of powers of a binomial, which is an expression consisting of two terms. The theorem states that:
(a+b)^n = a^n + na^(n-1)b + n(n-1)a^(n-2)b^2 + … + nab^(n-1) + b^n
where n is a positive integer.
Deriving the (a+b)^3 Formula
To derive the (a+b)^3 formula, we'll use the binomial theorem with n = 3. Substituting n = 3 into the theorem, we get:
(a+b)^3 = a^3 + 3a^2b + 3a^2b^2 + b^3
Now, let's analyze each term:
Term 1: a^3
The first term is simply a cubed, which is a^3.
Term 2: 3a^2b
The second term is 3 times a squared, multiplied by b. This is 3a^2b.
Term 3: 3ab^2
The third term is 3 times a, multiplied by b squared. This is 3ab^2.
Term 4: b^3
The fourth term is simply b cubed, which is b^3.
The Final Formula
Combining the four terms, we get the final formula for (a+b)^3:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
This formula allows us to expand the cube of a binomial expression into its individual terms.
Conclusion
In conclusion, the derivation of the (a+b)^3 formula is based on the binomial theorem. By using the theorem with n = 3, we can expand the cube of a binomial expression into four individual terms. This formula is essential in various mathematical applications, including algebra, geometry, and calculus.
