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Chapter 14: Problem 36

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

Short Answer

Expert verified
The first three terms of the binomial \((x^{2} + 1)^{17}\) expanded using the binomial theorem in simplified form are: \(x^{34} + 17x^{32} + 136x^{30}\).

Step by step solution

01

Understand the Binomial Theorem

The binomial theorem describes the algebraic expansion of powers of a binomial. The binomial theorem can be written as: \[(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\] where the symbol \({n \choose k}\) denotes a binomial coefficient.
02

Calculate First Three Terms

Using the binomial theorem, let's calculate the first three terms of the binomial. Here \(a = x^2\), \(b = 1\) and \(n=17\). The first term = \({17 \choose 0} *(x^{2})^{17-0} * 1^{0} = x^{34}\), second term = \({17 \choose 1} *(x^{2})^{17-1} * 1^{1} = 17*x^{32}\), and third term = \({17 \choose 2} *(x^{2})^{17-2} * 1^{2} = 136*x^{30}\).
03

Express the Result in Simplified Form

The first three terms of the binomial in simplified form are: \(x^{34} + 17x^{32} + 136x^{30}\).

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Most popular questions from this chapter

Factor: \(27 x^{3}-8\) (Section 6.4, Example 8)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Show that the sum of the first \(n\) positive odd integers, $$1+3+5+\cdots+(2 n-1)$$ is \(n^{2}\)

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$-1,1,-1,1, \dots$$

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$$

Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=4 n-3\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?
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