Use the Binomial Theorem to expand the binomial and express the result in simplified form. ((x^2)-1)^4

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Step 1: Recall the Binomial Theorem, which states that (a + b)^n can be expanded as Σ[k=0 to n] (n choose k) * a^(n-k) * b^k. Here, the binomial is ((x^2) - 1)^4, so a = x^2, b = -1, and n = 4.

Step 2: Write the general term for the expansion using the Binomial Theorem: T_k = (4 choose k) * (x^2)^(4-k) * (-1)^k, where k ranges from 0 to 4.

Step 3: Calculate each term of the expansion by substituting k = 0, 1, 2, 3, and 4 into the general term formula. For example, when k = 0, the term is (4 choose 0) * (x^2)^4 * (-1)^0.

Step 4: Simplify each term individually. For instance, (4 choose 0) = 1, (x^2)^4 = x^8, and (-1)^0 = 1, so the first term is x^8. Repeat this process for k = 1, 2, 3, and 4.

Step 5: Combine all the simplified terms to write the full expansion of ((x^2) - 1)^4 in simplified form.

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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 for expanding expressions of the form (a + b)^n, where n is a non-negative integer. It states that (a + b)^n can be expressed as the sum of terms in the form of C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient. This theorem is essential for systematically expanding binomials and is foundational in combinatorics.

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 without regard to the order of selection. They are calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. These coefficients play a crucial role in the expansion of binomials as they determine the coefficients of each term in the expanded form.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to understand and work with. This process may include combining like terms, factoring, and reducing fractions. In the context of the binomial expansion, simplification is necessary to present the final result clearly, ensuring that all terms are expressed in their lowest terms and combined appropriately.

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