Find each product. See Examples 5 and 6. (q-2)^4

Verified step by step guidance

1. Recognize that \((q-2)^4\) is a binomial raised to a power, which can be expanded using the Binomial Theorem.
2. The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = q\), \(b = -2\), and \(n = 4\).
3. Calculate each term of the expansion using the formula: \(\binom{4}{k} q^{4-k} (-2)^k\) for \(k = 0, 1, 2, 3, 4\).
4. Compute the binomial coefficients \(\binom{4}{k}\) for each term: \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \binom{4}{4}\).
5. Substitute the values into the expansion formula and simplify each term to get the expanded polynomial.

Key Concepts

Polynomial Expansion

Polynomial expansion involves expressing a polynomial raised to a power as a sum of terms. For example, using the binomial theorem, (a + b)^n can be expanded into a series of terms involving combinations of a and b. This concept is crucial for simplifying expressions like (q - 2)^4.

Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form (a + b)^n. It states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem is essential for calculating the coefficients and terms in the expansion of (q - 2)^4.

Coefficients and Combinations

In the context of polynomial expansion, coefficients are the numerical factors in front of the variable terms. The combinations, represented as 'n choose k', determine how many ways you can select k items from n, which directly influences the coefficients in the expanded polynomial. Understanding how to calculate these is vital for finding the product of (q - 2)^4.