Problem 36 - Algebra and Trigonometry

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(x^{2}+1\right)^{17} $$

Step-by-Step Solution

Answer

The first three terms in the expansion of \((x^{2}+1)^{17}\) are \(x^{34} + 17x^{32} + 136x^{30}\)

Step 1: Identify the binomial theorem

The binomial theorem states that for any real numbers a, b and for any positive integer n, \((a+b)^n = a^n + na^{n-1}b + \frac{n(n-1)}{2!}a^{n-2}b^2 + ··· + b^n\)

Step 2: Apply the theorem to the first term

The first term happens when all powers are taken by \(x^2\) and none by the other term, which is 1. Thus the first term is \((x^2)^{17} = x^{34}\)

Step 3: Calculate the second term

The second term happens when power 16 is taken by \(x^2\) and 1 power by the other term, which is 1. Therefore the second term is \(17x^{32}\)

Step 4: Determine the third term

In the third term, power 15 is taken by \(x^2\) and 2 powers by the other term, which is 1. Thus, the third term is \(\frac{17*16}{2!}x^{30} = 136x^{30}\)