Use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x − y)⁵

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Identify the binomial expression to expand: $(3x - y)^5$.

Recall the Binomial Theorem formula: $\displaystyle (a - b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} (-b)^k$.

Set $a = 3x$, $b = y$, and $n = 5$. Write the expansion as $\displaystyle \sum_{k=0}^5 \binom{5}{k} (3x)^{5-k} (-y)^k$.

Calculate each term by finding the binomial coefficient $\binom{5}{k}$, raising \$3x$ to the power \(5-k$, and \)-y$ to the power $k$, then multiply all parts together.

Simplify each term by applying the powers and combining like terms, paying attention to the signs from $(-y)^k$, and write the full expanded polynomial.

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Binomial Theorem

The Binomial Theorem provides a formula to expand expressions raised to a power, such as (a + b)^n. It states that the expansion is the sum of terms involving binomial coefficients, powers of a, and powers of b. This theorem simplifies the process of expanding binomials without multiplying repeatedly.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n. They appear as coefficients in the Binomial Theorem expansion and can be found using Pascal's Triangle or the formula n! / (k!(n-k)!). These coefficients determine the weight of each term in the expansion.

Exponent Rules in Expansion

When expanding a binomial like (3x − y)^5, each term involves powers of both 3x and −y. The exponents of these terms add up to the total power (5 in this case). Understanding how to apply exponent rules and distribute powers correctly is essential to simplify each term in the expansion.

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x − y)5