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Chapter 14: Problem 22

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$\left(2 x^{5}-1\right)^{4}$$

Short Answer

Expert verified
The simplified form of the \( \left(2x^{5} - 1\right)^{4} \) using the Binomial theorem is \( 16x^{20} - 32x^{15} + 24x^{10} -8x^{5} + 1 \)

Step by step solution

01

Identify the Binomial and its Power

The given expression is \( \left(2x^{5} - 1\right)^{4} \), where the binomial is \(2x^{5} - 1\) and the power is 4.
02

Apply the Binomial Theorem

The binomial theorem for any power \(n\) is given by \((a+b)^n = \sum_{k=0}^{n} {n \choose k}a^{n-k}b^{k}\). Here, \(a=2x^{5}\) and \(b=-1\). Applying the binomial theorem, we expand the given binomial as \((2x^{5} - 1)^{4} = \sum_{k=0}^{4} {4 \choose k}(2x^{5})^{4-k}(-1)^{k}\).
03

Expand using the Combination formula and Simplify

The expression becomes \({4 \choose 0}(2x^{5})^{4}(-1)^{0} + {4 \choose 1}(2x^{5})^{3}(-1)^{1} + {4 \choose 2}(2x^{5})^{2}(-1)^{2} + {4 \choose 3}(2x^{5})^{1}(-1)^{3} + {4 \choose 4}(2x^{5})^{0}(-1)^{4}\)which simplifies to \(16x^{20} - 32x^{15} + 24x^{10} -8x^{5} + 1\).

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Most popular questions from this chapter

What is a sequence? Give an example with your description.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. One of the terms in my binomial expansion is \(\left(\begin{array}{l}7 \\\ 5\end{array}\right) x^{2} y^{4}\).

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 69 and 70 to verify the expansion. $$f_{1}(x)=(x-2)^{4}$$

A deposit of 10,000 dollars is made in an account that earns \(8 \%\) interest compounded quarterly. The balance in the account after \(n\) quarters is given by the sequence $$a_{n}=10,000\left(1+\frac{0.08}{4}\right)^{n}, \quad n=1,2,3, \ldots$$ Find the balance in the account after six years. Round to the nearest cent.

$$\text { Solve: } \frac{6}{x}+\frac{6}{x+2}=\frac{5}{2}$$ (Section 7.6, Example 3)
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