Find each product. (a-b)(a^+ab+b^2)

Verified step by step guidance

1. Identify the expression to be expanded: \((a-b)(a^2+ab+b^2)\).
2. Apply the distributive property (also known as the FOIL method for binomials) to multiply each term in the first binomial \((a-b)\) by each term in the second trinomial \((a^2+ab+b^2)\).
3. Multiply \(a\) by each term in the trinomial: \(a \cdot a^2\), \(a \cdot ab\), and \(a \cdot b^2\).
4. Multiply \(-b\) by each term in the trinomial: \(-b \cdot a^2\), \(-b \cdot ab\), and \(-b \cdot b^2\).
5. Combine all the products from the previous steps and simplify by combining like terms, if any.

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In this case, recognizing the structure of the expression (a-b) and (a^2 + ab + b^2) is essential for simplifying the multiplication process.

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another polynomial. This requires applying the distributive property and combining like terms, which is crucial for finding the product of (a-b) and (a^2 + ab + b^2).

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. After multiplying the polynomials, it is important to identify and combine any like terms to arrive at the final simplified expression.