Find each product. See Example 5. (q - 2)⁴

Verified step by step guidance

1. Recognize that the expression \((q - 2)^4\) is a binomial raised to the fourth power. To expand it, you can use the Binomial Theorem, which states that \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\). In this case, \(a = q\), \(b = -2\), and \(n = 4\).
2. Write out the general expansion for \((q - 2)^4\) using the Binomial Theorem: \[ (q - 2)^4 = \binom{4}{0} q^4 (-2)^0 + \binom{4}{1} q^3 (-2)^1 + \binom{4}{2} q^2 (-2)^2 + \binom{4}{3} q^1 (-2)^3 + \binom{4}{4} q^0 (-2)^4 \]
3. Calculate each binomial coefficient \(\binom{4}{k}\) for \(k = 0, 1, 2, 3, 4\). These coefficients are the number of ways to choose \(k\) elements from 4 and can be found using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) or from Pascal's Triangle.
4. Evaluate the powers of \(q\) and \(-2\) for each term. For example, \(q^{4-k}\) and \((-2)^k\) for each \(k\) from 0 to 4.
5. Combine the coefficients, powers of \(q\), and powers of \(-2\) for each term to write the full expanded polynomial as a sum of terms. This will give you the expanded form of \((q - 2)^4\).

Key Concepts

Binomial Expansion

Binomial expansion is a method to expand 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 uses binomial coefficients from Pascal's triangle or combinations to determine each term's coefficient.

Binomial Theorem Formula

The binomial theorem states that (x + y)^n = Σ [n choose k] x^(n-k) y^k, where k ranges from 0 to n. This formula provides a systematic way to expand powers of binomials without multiplying repeatedly.

Binomial Coefficients and Pascal's Triangle

Binomial coefficients, denoted as C(n, k), represent the number of ways to choose k elements from n and serve as coefficients in binomial expansions. Pascal's triangle is a triangular array that lists these coefficients, simplifying the calculation of terms in expansions.