3-formula-proof.jpeg "(a+b)3 Formula Proof")
(a+b)^3 Formula Proof
The formula for (a+b)^3 is a fundamental concept in algebra, and it's essential to understand its proof to appreciate its beauty and importance.
The Formula
The formula for (a+b)^3 is:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
Proof
To prove this formula, we can use the concept of binomial expansion. The binomial theorem 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.
Case: n = 3
Let's apply the binomial theorem to the case where n = 3:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
To prove this, we can start with the left-hand side of the equation and expand it using the distributive property of multiplication over addition:
(a+b)(a+b)(a+b) = ?
Expanding the above expression, we get:
(a+b)(a^2 + 2ab + b^2) = a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3
Combine like terms:
a^3 + 3a^2b + 3ab^2 + b^3
which is the right-hand side of the equation.
Conclusion
Therefore, we have successfully proved the formula for (a+b)^3 using the binomial theorem. This formula is a powerful tool in algebra and is widely used in various mathematical and real-world applications.
