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 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, \(a = q\), \(b = -2\), and \(n = 4\).
3. Write out the expansion using the Binomial Theorem: \((q - 2)^4 = \sum_{k=0}^{4} \binom{4}{k} q^{4-k} (-2)^k\).
4. Calculate each term of the expansion: \(\binom{4}{0} q^4 (-2)^0\), \(\binom{4}{1} q^3 (-2)^1\), \(\binom{4}{2} q^2 (-2)^2\), \(\binom{4}{3} q^1 (-2)^3\), \(\binom{4}{4} q^0 (-2)^4\).
5. Combine all the terms to form the expanded expression.

Key Concepts

Polynomial Expansion

Polynomial expansion involves expressing a polynomial in a simplified form by multiplying out its factors. In this case, (q - 2)⁴ means multiplying (q - 2) by itself four times. Understanding how to expand polynomials is crucial for simplifying expressions and solving equations in algebra and trigonometry.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ. It states that (a + b)ⁿ can be expressed as a sum of terms involving binomial coefficients. For (q - 2)⁴, the theorem can be applied to find each term in the expansion, which is essential for calculating the product accurately.

Coefficients and Exponents

Coefficients are the numerical factors in terms of a polynomial, while exponents indicate the power to which a variable is raised. In the expansion of (q - 2)⁴, understanding how to determine the coefficients of each term and how exponents change during multiplication is vital for correctly finding the final expression.