Find each product. See Examples 5 and 6. (z-3)^3

Verified step by step guidance

1. Recognize that \((z-3)^3\) is a binomial raised to a power, which can be expanded using the Binomial Theorem.
2. The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \(a = z\), \(b = -3\), and \(n = 3\).
3. Write the expansion: \((z-3)^3 = \binom{3}{0} z^3 (-3)^0 + \binom{3}{1} z^2 (-3)^1 + \binom{3}{2} z^1 (-3)^2 + \binom{3}{3} z^0 (-3)^3\).
4. Calculate each binomial coefficient: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), \(\binom{3}{3} = 1\).
5. Substitute the coefficients and simplify each term: \(1 \cdot z^3 \cdot 1 + 3 \cdot z^2 \cdot (-3) + 3 \cdot z \cdot 9 + 1 \cdot 1 \cdot (-27)\).

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion is achieved using the Binomial Theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem allows for systematic calculation of each term in the expansion.

Cubic Functions

A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax^3 + bx^2 + cx + d. The graph of a cubic function can have one or two turning points and can exhibit various shapes, including inflection points. Understanding cubic functions is essential for analyzing their behavior and roots.

Factoring and Roots

Factoring involves breaking down a polynomial into simpler components, which can help in finding its roots or solutions. For a cubic expression like (z - 3)^3, recognizing that it represents a repeated root is crucial. The roots of the polynomial indicate where the function intersects the x-axis, providing insight into its behavior and solutions.