Write the first three terms in each binomial expansion, expressing the result in simplified form. (x² + 1)¹⁶

Verified step by step guidance

1. Identify the binomial expression and the exponent: here, the binomial is \((x^{2} + 1)\) and the exponent is \(16\).
2. Recall the Binomial Theorem formula for expansion: \(\displaystyle (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.
3. Write the first three terms by substituting \(k = 0, 1, 2\) into the formula: - For \(k=0\): \(\binom{16}{0} (x^{2})^{16} (1)^0\), - For \(k=1\): \(\binom{16}{1} (x^{2})^{15} (1)^1\), - For \(k=2\): \(\binom{16}{2} (x^{2})^{14} (1)^2\).
4. Simplify the powers of \(x\): - \((x^{2})^{16} = x^{32}\), - \((x^{2})^{15} = x^{30}\), - \((x^{2})^{14} = x^{28}\).
5. Express each term fully with coefficients and powers of \(x\): - \(\binom{16}{0} x^{32}\), - \(\binom{16}{1} x^{30}\), - \(\binom{16}{2} x^{28}\), and simplify the binomial coefficients.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n into a sum involving terms with coefficients, powers of a, and powers of b. Each term is given by the binomial coefficient multiplied by a^(n-k) and b^k, where k ranges from 0 to n.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n and serve as the coefficients in the binomial expansion. They can be calculated using factorials or found in Pascal's Triangle.

Simplifying Powers and Terms

After applying the binomial theorem, each term involves powers of the binomial components. Simplifying these powers, such as (x^2)^(n-k) = x^{2(n-k)}, and combining like terms ensures the expression is in its simplest form for clarity and further use.