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Chapter 14: Problem 31

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$(x+2)^{8}$$

Short Answer

Expert verified
The first three terms in the expansion of \((x+2)^8\) are \(x^{8} + 16x^{7} + 112x^{6}\).

Step by step solution

01

Understanding the Binomial Theorem

The Binomial Theorem characterizes the algebraic expansion of powers of a binomial. Given a binomial \((a + b)^n\), its expansion is given by the formula:\[(a + b)^n = \sum_{k=0}^{n} {n \choose k}a^{n-k}b^{k}\]where \(n \choose k\) refers to the binomial coefficient.
02

Expand the first three terms

In this exercise, \(a = x\), \(b = 2\), and \(n = 8\). Using the formula, the first three terms of the expansion will be:\[{x+2}^{8} = {8 \choose 0}x^{8-0}2^{0} + {8 \choose 1}x^{8-1}2^{1} + {8 \choose 2}x^{8-2}2^{2} \]Now, compute the binomial coefficients and simplify the exponents:
03

Simplify the expressions

Compute the binomial coefficients \(8 \choose 0\), \(8 \choose 1\), and \(8 \choose 2\), and simplify the resulting expressions:\[\begin{align*}{x+2}^{8} &= {8 \choose 0}x^{8}2^{0} + {8 \choose 1}x^{7}2^{1} + {8 \choose 2}x^{6}2^{2} \&= 1*x^{8}*1 + 8*x^{7}*2 + 28*x^{6}*4 \&= x^{8} + 16x^{7} + 112x^{6}\end{align*}\]

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Most popular questions from this chapter

Use the formula for the general term (the nth term) of a geometric sequence to solve. Suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. What will you put aside for savings on the thirtieth day of the month?

Use the formula for the sum of the first n terms of a geometric sequence to solve. You are investigating two employment opportunities. Company A offers \(\$ 30,000\) the first year. During the next four years, the salary is guaranteed to increase by \(6 \%\) per year. Company B offers \(\$ 32,000\) the first year, with guaranteed annual increases of \(3 \%\) per year after that. Which company offers the better total salary for a five-year contract? By how much? Round to the nearest dollar.

Expand and write the answer as a single logarithm with a coefficient of 1. $$\sum_{i=2}^{4} 2 i \log x$$

Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the exponents on \(a\).

Will help you prepare for the material covered in the next section. Use the formula \(a_{n}=a_{1} 3^{n-1}\) to find the 7 th term of the sequence \(11,33,99,297, \dots\)
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