(a%5E2%2Bab%2Bb%5E2)-(a%2Bb)(a%5E2-ab%2Bb%5E2)%3D-2b%5E3.jpeg "(a-b)(a^2+ab+b^2)-(a+b)(a^2-ab+b^2)=-2b^3")
Proving the Identity: (a-b)(a^2+ab+b^2)-(a+b)(a^2-ab+b^2)=-2b^3
This article aims to prove the algebraic identity: (a-b)(a^2+ab+b^2)-(a+b)(a^2-ab+b^2)=-2b^3
We will achieve this by expanding both sides of the equation and simplifying the results.
Expanding the Left-Hand Side
Let's begin by expanding the left-hand side of the equation: (a-b)(a^2+ab+b^2)-(a+b)(a^2-ab+b^2)
We can use the distributive property (also known as FOIL) to expand each term:
- (a-b)(a^2+ab+b^2) = a(a^2+ab+b^2) - b(a^2+ab+b^2)
- (a+b)(a^2-ab+b^2) = a(a^2-ab+b^2) + b(a^2-ab+b^2)
Expanding further, we get:
- a(a^2+ab+b^2) - b(a^2+ab+b^2) = a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3
- a(a^2-ab+b^2) + b(a^2-ab+b^2) = a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3
Combining like terms on both sides, we get:
- a^3 + a^2b + ab^2 - a^2b - ab^2 - b^3 = a^3 - b^3
- a^3 - a^2b + ab^2 + a^2b - ab^2 + b^3 = a^3 + b^3
Now, subtracting the second expression from the first:
(a^3 - b^3) - (a^3 + b^3) = -2b^3
Conclusion
Therefore, we have successfully proven the identity: (a-b)(a^2+ab+b^2)-(a+b)(a^2-ab+b^2)=-2b^3
This demonstrates that the expression on the left-hand side simplifies to the expression on the right-hand side, which is -2b^3.