Find each product. (r+3)⁴

Verified step by step guidance

1. Recognize that the expression \((r+3)^4\) 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: \n\n\[(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\]\n\nwhere \(\binom{n}{k}\) are the binomial coefficients.
3. Identify \(a = r\), \(b = 3\), and \(n = 4\). Then write out each term of the expansion using the formula: \n\n\(\binom{4}{0} r^4 3^0 + \binom{4}{1} r^3 3^1 + \binom{4}{2} r^2 3^2 + \binom{4}{3} r^1 3^3 + \binom{4}{4} r^0 3^4\)
4. Calculate each binomial coefficient \(\binom{4}{k}\) and write the terms explicitly: \n\n\$1 \cdot r^4 \cdot 1 + 4 \cdot r^3 \cdot 3 + 6 \cdot r^2 \cdot 9 + 4 \cdot r \cdot 27 + 1 \cdot 1 \cdot 81$
5. Multiply the coefficients and powers of 3 in each term, then combine all terms to write the fully expanded polynomial.

Key Concepts

Binomial Expansion

Binomial expansion is the process of expanding expressions raised to a power, such as (a + b)^n, into a sum of terms involving coefficients, powers of a, and powers of b. It allows us to write the expression as a polynomial without multiplying repeatedly.

Binomial Theorem and Coefficients

The Binomial Theorem provides a formula to expand (a + b)^n using binomial coefficients, which are found using combinations (n choose k). These coefficients determine the weight of each term in the expansion and can be found using Pascal's Triangle or the combination formula.

Powers of Variables and Constants

When expanding, each term involves powers of the variable and constant that add up to the exponent n. For (r + 3)^4, terms will include powers of r from 4 down to 0 and powers of 3 from 0 up to 4, multiplied by the corresponding binomial coefficient.