Use the Binomial Theorem to expand each binomial and express the result in simplified form. (y-3)⁴

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Recall the Binomial Theorem formula: \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \), where \(\binom{n}{k}\) is the binomial coefficient.

Identify the terms in the binomial: here, \(a = y\), \(b = -3\), and \(n = 4\).

Write the expansion as \( \sum_{k=0}^{4} \binom{4}{k} y^{4-k} (-3)^k \).

Calculate each term separately for \(k = 0, 1, 2, 3, 4\) by finding the binomial coefficients \(\binom{4}{k}\), powers of \(y\), and powers of \(-3\).

Combine all terms and simplify the expression by multiplying coefficients and powers to get the expanded form.

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주요 개념

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Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n equals the sum of terms C(n, k) * a^(n-k) * b^k, where C(n, k) are binomial coefficients. This theorem simplifies the expansion process without multiplying the binomial repeatedly.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from a set of n elements. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal's Triangle. These coefficients determine the weight of each term in the expansion.

Simplifying Algebraic Expressions

After expanding a binomial using the Binomial Theorem, simplifying involves combining like terms and performing arithmetic operations. This step ensures the final expression is in its simplest form, making it easier to interpret or use in further calculations. Simplification includes applying exponent rules and arithmetic with constants.

Use the Binomial Theorem to expand each binomial and express the result in simplified form. (y-3)4