In Exercises 15–58, find each product. (x+2)^3

Key Concepts

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly those in the form of (a + b)^n. The expansion can be systematically 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 provides a formula for calculating the coefficients of the expanded terms.

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. In the context of the question, (x + 2)^3 represents a cubic function where the variable x is transformed by adding 2 before being cubed. Understanding cubic functions is essential for recognizing their properties, such as their shape and the behavior of their graphs.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials to produce a new polynomial. This process requires distributing each term in the first polynomial to every term in the second polynomial, often using the distributive property or the FOIL method for binomials. Mastery of polynomial multiplication is crucial for simplifying expressions and solving algebraic equations.