Problem 35 - Algebra and Trigonometry

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(x^{2}+1\right)^{16} $$

Step-by-Step Solution

Answer

The first three terms of the binomial expansion \( (x^{2}+1)^{16} \) are: \(x^{32}, 16 * x^{30}, 120 *x^{28}\).

Step 1: Understand the Binomial Theorem

Before diving into the simplification, we need to understand the Binomial Theorem. In general, when expanding the expression \( (a + b)^n \), the theorem can be expressed as \( (a+ b)^n = a^n + nab^{n-1} + {n(n-1)}a^{n-2}b^2 /2! + ... + b^n \), where n is a positive integer. On the nth term, the coefficient can be written as (n choose k).

Step 2: Apply the Binomial Theorem

Applying the Binomial Theorem to the given problem \( (x^{2} + 1)^{16} \). This can be rewritten as the expansion: \( (x^2)^{16} + {16 \choose 1} (x^2)^{15} + {16 \choose 2} (x^2)^{14} + ... \).

Step 3: Simplify the first three terms

(When k = 0) First term = \((x^2)^ {16}\) = \(x^{32}\). (When k = 1) Second term = {16 \choose 1} * \((x^2)^ {15}\) = 16*\(x^{30}\). (When k = 2) Third term = {16 \choose 2} * \((x^2)^ {14}\) = 120*\(x^{28}\). Hence, the first three terms will be \(x^{32}, 16 * x^{30}, 120 *x^{28}\).