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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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 the expansion can be expressed as a sum of terms involving binomial coefficients, which are calculated using the formula C(n, k) = n! / (k!(n-k)!), where k ranges from 0 to n. This theorem is essential for simplifying expressions involving powers of binomials.

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Binomial Coefficients

Binomial coefficients are the numerical factors that appear in the expansion of a binomial expression according to the Binomial Theorem. They represent the number of ways to choose k elements from a set of n elements and are denoted as C(n, k) or 'n choose k'. These coefficients play a crucial role in determining the coefficients of each term in the expanded form of the binomial expression.

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Simplification of Expressions

Simplification of expressions involves combining like terms and reducing expressions to their simplest form. In the context of binomial expansions, this means collecting terms with the same variable powers and constants. This process is important for making the final result more manageable and easier to interpret, especially when dealing with higher powers and multiple terms.

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