Prove: a3+b3 Formula

Contents

$a^{3} + b^{3 formula}$ is a cornerstone in algebra, serving as a critical tool for simplifying and solving cubic expressions. Its relevance spans various mathematical disciplines, from basic algebra to advanced calculus. Understanding this formula not only enhances problem-solving skills but also lays the groundwork for tackling more complex algebraic challenges. The formula is expressed as:

$a3+b3 = (a + b) (a^{2} - ab + b^{2})$

This article delves into a detailed proof of this formula, illuminating its derivation and practical significance.

Proving The a3+b3 Formula

The proof of $a^{3} + b^{3}$ hinges on demonstrating that the expanded and simplified form of the right-hand side (RHS) of the equation is equal to the left-hand side (LHS).

Step 1: Expand the Right-Hand Side

First, we focus on expanding the RHS of the equation, $(a + b) (a^{2} - ab + b^{2})$ . This involves applying the distributive property of multiplication over addition.

$(a + b) (a^{2} - ab + b^{2}) = a (a^{2} - ab + b^{2}) + b (a^{2} - ab + b^{2})$

This expansion is the first critical step in unraveling the formula.

Step 2: Distribute the Terms

Next, we distribute the terms and within their respective brackets. This step is crucial as it lays the foundation for the subsequent simplification.

$=a^{3} - a^{2} b + a b^{2} + b a^{2} - b ab + b^{3}$

Each term is carefully multiplied, ensuring that all possible combinations are accounted for.

Step 3: Simplify the Expression

In this step, we simplify by combining like terms. It’s important to identify and cancel out terms that negate each other.

$=a^{3} - a^{2} b + a b^{2} + a^{2} b - a b^{2} + b^{3}$

Notice how $-a^{2} b$ and $+ a^{2} b$ cancel each other, as do $a b^{2}$ and $- a b^{2}$ . This cancellation is a pivotal part of the proof.

Step 4: Final Simplification

After the cancellation, the equation simplifies to:

$=a^{3} + b^{3}$

This final step confirms that the RHS of our original formula simplifies to match the LHS exactly.

Must Read:

Conclusion

Through systematic expansion, distribution, and simplification of a3+b3, we have shown that a3+b3 $= (a + b) (a^{2} - ab + b^{2})$ . This proof not only validates the formula but also highlights the elegance of algebraic manipulation. The ability to factorize and simplify cubic expressions using this formula is a valuable skill in mathematics.

Prove: a3+b3 Formula