/*! 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 15 Use the Binomial Theorem to expa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 15

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

Short Answer

Expert verified
The simplified form of the binomial expression \((2x+1)^4\) is \(16x^4 + 32x^3 + 24x^2 + 8x + 1\).

Step by step solution

01

Using the Binomial Theorem

Start by writing the expression where Binomial Theorem can be applied. Here, Binomial Theorem states that \((a+b)^n=\sum_{k=0}^{n}\) \(n\choose k\) \(a^{n-k} b^{k}\) where \(n\choose k\) is the binomial coefficient. So \((2x+1)^4=\sum_{k=0}^{4}\) \(4\choose k\) \((2x)^{4-k} 1^{k}\)
02

Expanding Through the Binomial Coefficients

We then apply the binomial coefficients (binoms) from \(k=0\) to \(k=4\). These are calculated as \(4\choose 0\), \(4\choose 1\), \(4\choose 2\), \(4\choose 3\), and \(4\choose 4\) respectively. Therefore, the expanded form is \(4\choose 0\) \((2x)^4\cdot 1^0\) + \(4\choose 1\)\((2x)^3\cdot 1^1\) + \(4\choose 2\)\((2x)^2\cdot 1^2\) + \(4\choose 3\)\((2x)^1\cdot 1^3\) + \(4\choose 4\)\((2x)^0\cdot 1^4\)
03

Simplifying the Expression

Now simplify the expression by calculating each term. This will give \(1\cdot16x^4\) + \(4\cdot8x^3\) + \(6\cdot4x^2\) + \(4\cdot2x\) + \(1\cdot1\). After simplifying, it yields \(16x^4 + 32x^3 + 24x^2 + 8x + 1\).

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

What is an annuity?

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$$\begin{array}{l}2+2\left(\frac{1}{3}\right)+2\left(\frac{1}{3}\right)^{2} \\\\+2\left(\frac{1}{3}\right)^{3}+\cdots\end{array}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=1}^{4} 3 i+\sum_{i=1}^{4} 4 i=\sum_{i=1}^{4} 7 i$$

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. An arithmetic sequence is a linear function whose domain is the set of natural numbers
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.