/*! 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 83 Find the term in the expansion o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 83

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

Short Answer

Expert verified
The term in the expansion \((x^{2} + y^{2})^{5}\) containing \(x^{4}\) as a factor is \(10x^{4}y^{6}\).

Step by step solution

01

Understanding the binomial expansion

The binomial theorem describes the algebraic expansion of powers of a binomial. According to the binomial theorem, any power of the binomial sum \(a+b\) can be expressed as: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^{k}\), where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) is the binomial coefficient.
02

Identify the coefficients

In \((x^{2}+y^{2})^{5}\) expansion, \(a = x^2, b = y^2\), and \(n = 5\). We are looking for the term where \(x^4\) exists, so \(a\) should be raised to the 2nd power as \(x^{2*2}=x^{4}\), meaning \(k = 2\).
03

Calculate the coefficient of the term

The coefficient for the term is calculated using the binomial coefficient formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n = 5\) and \(k = 2\). It results in the value \(\binom{5}{2} = 10\). This is the coefficient in front of the \(x^{4}\).
04

Examine the y terms

The term \(y^{2}\) should be to the power of \(5 - 2 = 3\) to make the total power 5. So, the term in the expansion containing \(x^{4}\) as a factor is \(10x^{4}y^{6}\).

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

Use the formula for the sum of the first n terms of a geometric sequence to solve. You save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, continuing to double your savings each day. What will your total savings be for the first 15 days?

Will help you prepare for the material covered in the next section. Consider the sequence \(1,-2,4,-8,16, \ldots\) Find \(\frac{a_{2}}{a_{1}}, \frac{a_{3}}{a_{2}}, \frac{a_{4}}{a_{3}},\) and \(\frac{a_{5}}{a_{4}} .\) What do you observe?

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 69 and 70 to verify the expansion. $$f_{1}(x)=(x-2)^{4}$$

Will help you prepare for the material covered in the next section. Consider the sequence \(8,3,-2,-7,-12, \ldots .\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$1,4,9,16, \dots$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.