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

Verified step by step guidance

1. Recognize that the expression \((a - b)(a^2 + ab + b^2)\) matches the form of the difference of cubes factorization, where \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
2. Understand that multiplying \((a - b)\) by \((a^2 + ab + b^2)\) will give you the expanded form of \(a^3 - b^3\).
3. To verify, apply the distributive property (also known as FOIL for binomials) by multiplying each term in the first parenthesis by each term in the second parenthesis:
4. Calculate \(a \times a^2 = a^3\), \(a \times ab = a^2b\), \(a \times b^2 = ab^2\), \(-b \times a^2 = -a^2b\), \(-b \times ab = -ab^2\), and \(-b \times b^2 = -b^3\).
5. Combine like terms: notice that \(a^2b\) and \(-a^2b\) cancel out, as do \(ab^2\) and \(-ab^2\), leaving you with \(a^3 - b^3\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by every term in the other polynomial. This process requires applying the distributive property to combine like terms and simplify the expression.

Difference of Cubes Formula

The difference of cubes formula states that (a - b)(a^2 + ab + b^2) equals a^3 - b^3. Recognizing this pattern allows for quick simplification of expressions matching this form without performing full multiplication.

Exponents and Like Terms

Understanding exponents is essential for correctly interpreting terms like a^2 and b^2. Combining like terms involves adding or subtracting coefficients of terms with the same variables and exponents to simplify the final expression.