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Chapter 14: Problem 23

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(c+2)^{5}$$

Short Answer

Expert verified
The expanded and simplified form of \( (c+2)^5 \) is \( 32 + 160c + 320c^2 + 160c^3 + 10c^4 + c^5 \)

Step by step solution

01

Understand The Binomial Theorem

The Binomial Theorem follows the pattern \((a+b)^n = \sum_{k=0}^{n} {n\choose k} a^k b^{n-k}\). Which means it has n+1 terms in expanded form, and we can find the coefficient of each term using combinations. The variables have powers that increase for one variable and decrease for the other. The sum of the powers in any term is always equal to n.
02

Apply The Binomial Theorem To expand (c+2)^5

With \( (c + 2)^5 \), we have \( a=c, b=2, n=5 \). Now we can apply the theorem: \((c+2)^5 = \sum_{k=0}^{5} {5\choose k} c^k \cdot 2^{5-k}\).
03

Compute Each Term

After calculating each term, we get 32c^0 + 80c^1 + 80c^2 + 40c^3 + 10c^4 + c^5, which simplifies to 32 + 160c + 320c^2 + 160c^3 + 10c^4 + c^5.
04

Final Result

The fully expanded and simplified form of \( (c+2)^5 \) is \( 32 + 160c + 320c^2 + 160c^3 + 10c^4 + c^5 \)

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I've noticed that the big difference between arithmetic and geometric sequences is that arithmetic sequences are based on addition and geometric sequences are based on multiplication.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of \(f\) and discuss its relationship to the sum of the given series. Function $$f(x)=\frac{2\left[1-\left(\frac{1}{3}\right)^{x}\right]}{1-\frac{1}{3}}$$ Series$$\begin{array}{l}2+2\left(\frac{1}{3}\right)+2\left(\frac{1}{3}\right)^{2} \\\\+2\left(\frac{1}{3}\right)^{3}+\cdots\end{array}$$

If \(f(x)=x^{2}+2 x+3,\) find \(f(a+1)\) (Section \(8.1,\) Example 3 )

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$5,7,9,11, \dots$$
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