/*! 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 64 Explain how to use the Binomial ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 64

Explain how to use the Binomial Theorem to expand a binomial. Provide an example with your explanation.

Short Answer

Expert verified
The expanded form of \((x+y)^3\) is \(x^3 + 3x^2y + 3xy^2 + y^3\)

Step by step solution

01

Understanding the Binomial Theorem

The Binomial Theorem is given by: \((a + b)^n = \sum _{k=0} ^n \binom{n}{k} a^{n-k} b^{k}\). Here, \(a\) and \(b\) are the terms of the binomial and \(n\) is the power the binomial is raised to. \(\binom{n}{k}\) are the binomial coefficients, which can be found using the combination formula or Pascal's triangle.
02

Applying the Binomial Theorem for Expansion

Let's see an example by expanding \( (x+y)^3 \). According to the binomial theorem, this is equal to \( \binom{3}{0} x^3 y^0 + \binom{3}{1} x^2 y^1 + \binom{3}{2} x^1 y^2 + \binom{3}{3} x^0 y^3 \)
03

Calculating Binomial Coefficients

To find the coefficients, we can use the combination formula: \(\binom{n}{k}= \frac{n!}{k!(n-k)!}\), where '!' denotes a factorial. In our case, your calculations would be: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), and \(\binom{3}{3} = 1\)
04

Applying Binomial Coefficients

Now replace the binomial coefficients in your previous equation. Your final expanded binomial would be \(x^3 + 3x^2y + 3xy^2 + y^3\)
05

Confirm the Solution

Make sure all terms and coefficients align correctly with the rules of the binomial theorem. When you expand a binomial of power \(n\), you should get \(n + 1\) terms in the expansion. Also, each term's power must sum up to \(n\). In our example, we have 3 + 1 = 4 terms and each term's power sums up to 3 which verifies our solution.

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

Convert the equation $$ 4 x^{2}+y^{2}-24 x+6 y+9=0 $$ to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of the foci. (Section \(10.1,\) Example 5 )

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.

Use the formula for the general term (the nth term) of a geometric sequence to solve Exercises \(65-68\) In Exercises \(65-66,\) suppose you save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, and so on. That is, each day you save twice as much as you did the day before. You are offered a job that pays \(\$ 30,000\) for the first year with an annual increase of \(5 \%\) per year beginning in the second year. That is, beginning in year \(2,\) your salary will be 1.05 times what it was in the previous year. What can you expect to earn in your sixth year on the job?

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

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{M M M, M M F, M F M, M F F, F M M FMF, FFM, FFF\)} - Find the probability of selecting a family with $$\text{at least two female children.}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.