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Chapter 11: Problem 24

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (c+3)^{5} $$

Short Answer

Expert verified
\(c^{5} + 15c^{4} + 90c^{3} + 270c^{2} + 405c + 243\)

Step by step solution

01

Understand Binomial Theorem

The Binomial Theorem states that \((x+y)^{n}= \sum_{r=0}^{n} \binom{n}{r} x^{n-r} y^{r}\) where \(n\) is a nonnegative integer, \(x\) and \(y\) are variables, and \(\binom{n}{r}\) is the number of ways to choose \(r\) items from \(n\) items also known as combinations or 'n choose r'.
02

Expand the binomial using the theorem

In \((c+3)^{5}\), we have \(x=c\), \(y=3\) and \(n=5\). The expansion is as follows: \((c+3)^{5} = \binom{5}{0}c^{5}3^0 + \binom{5}{1}c^{4}3^1 + \binom{5}{2}c^{3}3^2 + \binom{5}{3}c^{2}3^3 + \binom{5}{4}c^{1}3^4 + \binom{5}{5}c^{0}3^5\).
03

Simplify Each Term

Now we simplify each term. The combinations \(\binom{5}{r}\) simplify as follows: \(\binom{5}{0}=1\), \(\binom{5}{1}=5\), \(\binom{5}{2}=10\), \(\binom{5}{3}=10\), \(\binom{5}{4}=5\), and \(\binom{5}{5}=1\). The powers of \(3\) simplify as: \(3^{0}=1, 3^{1}=3, 3^{2}=9, 3^{3}=27, 3^{4}=81\), and \(3^{5}=243\). Substituting these number into the expanded form gives \(1*c^{5}*1 + 5*c^{4}*3 + 10*c^{3}*9 + 10*c^{2}*27 + 5*c^{1}*81 + 1*c^{0}*243\).
04

Rewrite The Terms

To obtain simplified form for each term, we multiply the numbers together. We get \(c^{5} + 15c^{4} + 90c^{3} + 270c^{2} + 405c + 243\). This is our final answer.

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Most popular questions from this chapter

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

Solve triangle \(A B C\) if \(a=17, b=28,\) and \(c=15\). Round angle measures to the nearest degree.

What is the difference between a geometric sequence and an infinite geometric series?

Explaining the Concepts Give an example of two events that are not mutually exclusive.

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