Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x − 1)⁵

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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, \(a = x\), \(b = -1\), and \(n = 5\).

Write the expansion using the formula: \( (x - 1)^5 = \sum_{k=0}^5 \binom{5}{k} x^{5-k} (-1)^k \).

Calculate each term by finding \(\binom{5}{k}\), raising \(x\) to the power \(5-k\), and \((-1)\) to the power \(k\), then multiply these values together for each \(k\) from 0 to 5.

Combine all the terms from \(k=0\) to \(k=5\) to write the full expanded expression, and simplify the signs and coefficients where possible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 Polynomial Expressions

After expanding a binomial using the Binomial Theorem, simplifying involves combining like terms and reducing powers where possible. This step ensures the final expression is in its simplest form, making it easier to interpret or use in further calculations. Simplification also includes applying negative signs correctly when terms involve subtraction.

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