/*! 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 56 How do you determine how many te... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 56

How do you determine how many terms there are in a binomial expansion?

Short Answer

Expert verified
To determine the number of terms in a binomial expansion, simply add 1 to the exponent of the binomial. For example, in the binomial expansion of \((a+b)^n\), there are \(n+1\) terms.

Step by step solution

01

Understanding Binomial Expansion

In a binomial expansion, each term is formed using the binomial coefficients, whose values are determined by the binomial theorem, and the powers of the terms in the original binomial expression. A binomial expansion is the expansion of a power of a binomial such as \((a+b)^n\).
02

Applying Binomial Theorem

Based on the binomial theorem, the binomial expression \((a+b)^n\) is expanded as \(a^n + \left( ^nC_1 \right) a^{n-1}b + \left( ^nC_2 \right) a^{n-2}b^2 + \ldots + b^n\). Each coefficient in the expansion is a binomial coefficient, as found in the n-th row of Pascal's triangle or computed using the combination formula.
03

Determining the Number of Terms

In an expanded form, the number of terms is one more than the exponent on the binomial expression. This is because the first term is the nth power of the first term in the binomial, the second term is the (n-1)th power of the first term times the first power of the second term, and so on, until the last term which is the nth power of the second term in the binomial.

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

Evaluate the given binomial coefficient. $$ \left(\begin{array}{l}8 \\\3\end{array}\right) $$

Find the term indicated in each expansion. \((2 x+y)^{6} ;\) third term

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(x^{2}+1\right)^{17} $$

What is the difference between a geometric sequence and an infinite geometric series?

Graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. $$ \begin{array}{l}f_{1}(x)=(x+1)^{4} \\\f_{2}(x)=x^{4} \\\f_{3}(x)=x^{4}+4 x^{3} \\\f_{4}(x)=x^{4}+4 x^{3}+6 x^{2} \\\f_{5}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x \\\f_{6}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x+1\end{array} $$ Use a \([-5,5,1]\) by \([-30,30,10]\) viewing rectangle.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.