Find each product. (z-3)³

Verified step by step guidance

1. Recognize that the expression \((z-3)^3\) is a cube of a binomial, which means it can be expanded using the formula for the cube of a binomial: \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\).
2. Identify the terms in the binomial: here, \(a = z\) and \(b = 3\).
3. Apply the formula by substituting \(a = z\) and \(b = 3\) into each term: \(a^3 = z^3\), \(3a^2b = 3 \times z^2 \times 3\), \(3ab^2 = 3 \times z \times 3^2\), and \(b^3 = 3^3\).
4. Write out the expanded form using these substituted terms: \(z^3 - 3 \times z^2 \times 3 + 3 \times z \times 3^2 - 3^3\).
5. Simplify each term by performing the multiplications (but do not combine like terms yet if you want to keep the expression clear) to get the fully expanded polynomial.

Key Concepts

Binomial Expansion

Binomial expansion is the process of expanding expressions raised to a power, such as (a + b)^n, into a sum involving terms of the form a^k * b^(n-k). It uses the Binomial Theorem or Pascal's Triangle to find coefficients and powers systematically.

Exponentiation of Binomials

Exponentiation of binomials involves multiplying a binomial by itself multiple times, such as (z - 3)^3 = (z - 3)(z - 3)(z - 3). Understanding how to perform repeated multiplication and combine like terms is essential to simplify the expression.

Combining Like Terms

Combining like terms means adding or subtracting terms with the same variable raised to the same power. After expanding, grouping these terms simplifies the expression into a standard polynomial form.