/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Use the Binomial Theorem to expa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: 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 expanded form of \( (2x^{5}-1)^{4} \) using the binomial theorem is \(16x^{20} - 32x^{15} + 24x^{10} - 8x^{5} + 1\).

Step by step solution

01

Identify the components of the binomial

In the term \( (2x^{5}-1)^{4} \), \(a = 2x^{5}\), \(b = -1\) and \(n = 4\). These values will be used in the binomial theorem formula.
02

Use the binomial theorem to expand the binomial

Now proceed to the formula for the binomial theorem: \( (a+b)^{n} = 危 \binom{n}{k} a^{n-k} b^{k} \) for k=0 to n. After substitution, we get \( (2x^{5}-1)^{4} = 危 \binom{4}{k} (2x^{5})^{4-k} (-1)^{k} \) for k=0 to 4
03

Calculate each term of the result

Calculate the terms of the result, remembering that the binomial coefficient \( \binom{n}{k} \) is calculated as \( \frac{n!}{k!(n-k)!} \). Using these formulas, we get the terms: \n- When k=0, term = \( \binom{4}{0} (2x^{5})^{4} (-1)^{0} = 16x^{20} \)\n- When k=1, term = \( \binom{4}{1} (2x^{5})^{3} (-1)^{1} = -32x^{15} \)\n- When k=2, term = \( \binom{4}{2} (2x^{5})^{2} (-1)^{2} = 24x^{10} \)\n- When k=3, term = \( \binom{4}{3} (2x^{5})^{1} (-1)^{3} = -8x^{5} \)\n- When k=4, term = \( \binom{4}{4} (2x^{5})^{0} (-1)^{4} = 1 \)
04

Write down the final result

By adding up all terms computed above, we obtain the expanded expression: \(16x^{20} - 32x^{15} + 24x^{10} - 8x^{5} + 1\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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

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.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Assuming the next U.S. president will be a Democrat or a Republican, the probability of a Republican president is 0.5

The table shows the population of California for 2000 and \(2010,\) with estimates given by the U.S. Census Bureau for 2001 through 2009 \(\begin{array}{lllllll}\hline \text { Year } & {2000} & {2001} & {2002} & {2003} & {2004} & {2005} \\ \hline \text { Population } & {33.87} & {34.21} & {34.55} & {34.90} & {35.25} & {35.60} \\ \hline\end{array}\) \(\begin{array}{llllll}{\text { Year }} & {2006} & {2007} & {2008} & {2009} & {2010} \\ {\text { Population }} & {36.00} & {36.36} & {36.72} & {37.09} & {37.25}\end{array}\) a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after 1999 c. Use your model from part (b) to project California's population, in millions, for the year \(2020 .\) Round to two decimal places.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.