In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x² + 1)^16

Verified step by step guidance

1. Step 1: Identify the binomial expression and the power. Here, the binomial is \((x^2 + 1)\) and the power is 16.
2. Step 2: Use the Binomial Theorem, which states \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), to expand the expression.
3. Step 3: For the first term, set \(k = 0\). Calculate \(\binom{16}{0} (x^2)^{16-0} (1)^0\).
4. Step 4: For the second term, set \(k = 1\). Calculate \(\binom{16}{1} (x^2)^{16-1} (1)^1\).
5. Step 5: For the third term, set \(k = 2\). Calculate \(\binom{16}{2} (x^2)^{16-2} (1)^2\).

Key Concepts

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 determining the coefficients and terms in the expansion.

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 can be calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. These coefficients are crucial for finding the specific terms in a binomial expansion.

Simplification of Expressions

Simplification involves reducing expressions to their simplest form, which often includes combining like terms and eliminating unnecessary components. In the context of binomial expansions, this means expressing the first three terms clearly and concisely, ensuring that any powers or coefficients are correctly calculated and presented. This step is vital for clarity and ease of understanding.