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Chapter 14: Problem 21

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$\left(2 x^{3}-1\right)^{4}$$

Short Answer

Expert verified
The expanded form of the binomial \(\left(2x^{3}-1\right)^{4}\) is \(16x^{12} - 32x^{9} + 24x^{6} - 8x^{3} + 1\).

Step by step solution

01

Recall the Binomial Theorem

The Binomial Theorem can be written as: \((a+b)^{n}= \sum_ {k=0}^{n} {n \choose k} a^{n-k} b^{k}\), where \({n \choose k}\) is a binomial coefficient that can be calculated as \( \frac{n!}{k!(n-k)!}\), where \(n!\) is the factorial of n.
02

Identify the components and apply the theorem

In \((2x^{3}-1)^{4}\), \(a=2x^{3}\), \(b=-1\), and \(n=4\). Replace \(a\), \(b\), and \(n\) in the Binomial Theorem to calculate each term.
03

Calculate the coefficients and simplify

Evaluate \({n \choose k}\) for each \(k\) from 0 to 4, multiply each term by the corresponding power of \(a\) and \(b\), and simplify each term.
04

Sum all terms

Sum all the calculated terms to obtain the final expanded form of the given binomial.

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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 \(a_{n}=4 n-3\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

Use the formula for the value of an annuity to solve Exercises. Round answers to the nearest dollar. At age \(25,\) to save for retirement, you decide to deposit \(\$ 50\) at the end of each month in an IRA that pays \(5.5 \%\) compounded monthly. a. How much will you have from the IRA when you retire at age \(65 ?\) b. Find the interest.

Solve for \(P: A=\frac{P t}{P+t}\)

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Find the term in the expansion of \(\left(x^{2}+y^{2}\right)^{5}\) containing \(x^{4}\) as a factor.
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