Problem 35 Write the first three terms in e... [FREE SOLUTION]

Chapter 11: Problem 35

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

Short Answer

Expert verified

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

Step by step solution

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).

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} + ... $.

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 $x^{32}, 16 * x^{30}, 120 *x^{28}$.

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91影视

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

Short Answer

Step by step solution

Understand the Binomial Theorem

Apply the Binomial Theorem

Simplify the first three terms

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