Find each product. (q-2)⁴

Verified step by step guidance

1. Recognize that the expression \((q-2)^4\) is a binomial raised to the fourth power, which means you need to expand it using the Binomial Theorem or by repeated multiplication.
2. Recall the Binomial Theorem formula: \( (a - b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} (-b)^k \), where \(a = q\), \(b = 2\), and \(n = 4\).
3. Calculate the binomial coefficients \(\binom{4}{k}\) for \(k = 0, 1, 2, 3, 4\). These coefficients are 1, 4, 6, 4, and 1 respectively.
4. Write out each term of the expansion using the formula: \( \binom{4}{k} q^{4-k} (-2)^k \) for each \(k\) from 0 to 4.
5. Combine all terms to express the expanded polynomial: \( q^4 - 4q^3 \cdot 2 + 6q^2 \cdot 4 - 4q \cdot 8 + 1 \cdot 16 \), then simplify the coefficients (do not calculate the final numeric values here).

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 C(n, k) * a^(n-k) * b^k. It allows us to write powers of binomials as polynomials without multiplying repeatedly.

Binomial Theorem and Binomial Coefficients

The Binomial Theorem provides a formula to expand (a + b)^n using binomial coefficients, which are combinations represented as C(n, k) = n! / (k!(n-k)!). These coefficients determine the weights of each term in the expansion and can be found using Pascal's Triangle.

Polynomial Multiplication and Powers

Raising a binomial to a power involves multiplying the polynomial by itself multiple times. Understanding how to multiply polynomials and combine like terms is essential to simplify the expanded expression correctly.