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Chapter 14: 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 of \((x^{2}+1)^{16}\) are \(x^{32}\), \(16x^{30}\), and \(120x^{28}\).

Step by step solution

01

Determine the first term

The first term is the \(a^n\) where \(a=x^2\) and \(n=16\). So the first term is \( (x^2)^{16} = x^{32}\).
02

Determine the second term

The second term is \({n \choose 1} a^{n-1}b\), where \(n = 16\), \(a=x^2\), and \(b=1\). So the second term is \( {16 \choose 1} (x^2)^{15} * 1 = 16x^{30}\).
03

Determine the third term

The third term is \({n \choose 2} a^{n-2}b^2\), where \(n = 16\), \(a=x^2\), and \(b=1\). So the third term is \({16 \choose 2} (x^2)^{14} * 1^2 = 120x^{28}\).

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