/*! 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 45 Find the specified \(n\) th term... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 45

Find the specified \(n\) th term in the expansion of the binomial.$$(x+y)^{10}, \quad n=4$$.

Short Answer

Expert verified
The 4th term in the binomial expansion of \((x+y)^{10}\) is \(120 x^{7} y^{3}\).

Step by step solution

01

Identify the Parameters

In this exercise, the parameters are \(n=10\) indicating the power of the binomial expression and \(k=3\) (as the term index starts from 0, 4th term corresponds to index 3) as the term number.
02

Apply the Binomial Theorem

Substitute the values of \(n\) and \(k\) into the binomial theorem. The \(k\)th term of the binomial expansion \((x+y)^n\) can be found using the formula \(\binom{n}{k} x^{n-k} y^k\). Here, term = \(\binom{10}{3} x^{10-3} y^3\).
03

Calculate the Binomial Coefficient

The binomial coefficient, \(\binom{10}{3}\), can be calculated as 10! / (3!(10-3)!). This equals to 120.
04

Substitute Coefficient back into the expression

Put the calculated value of the binomial coefficient back in the equation: term = \(120 x^{7} y^{3}\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Binomial Expansion
Let's start by understanding what binomial expansion is. The binomial theorem provides a formula that helps us expand expressions that are raised to a power, like \[ (x + y)^n \]This means multiplying \[ (x + y) \] by itself **n** times.
  • The binomial expansion will result in terms where each term is a product of two parts: one part involving 'x' and the another part involving 'y'.
  • The sum of the exponents of 'x' and 'y' in each term equals the power **n** of the original binomial.
To find an exact term in such an expansion, we use the formula \[ \binom{n}{k} x^{n-k} y^k \]where **n** is the power and **k** is the term index starting from 0. It helps break down the expansion into a sequence of terms, with each individual term in the expansion represented by a binomial coefficient.
Introduction to Binomial Coefficient
The binomial coefficient, denoted as \[ \binom{n}{k} \], is a central concept in binomial expansion. It represents the number of ways to select 'k' elements from a set of 'n' elements without considering the order.
  • This coefficient can be derived using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
  • It is also known as "n choose k" and provides the combinatoric aspect of binomial expansions.
  • For example, in our exercise, \[ \binom{10}{3} \] calculates to 120. This means there are 120 possible ways to select 3 elements from 10, contributing to the coefficient of the specific term in the expansion.
Understanding binomial coefficients helps us determine the weight or coefficient of each term in binomial expansion.
Mastering Exponents in Binomial Expansion
Exponents play a significant role in binomial expansions. Each term in the expansion will carry an exponent on both 'x' and 'y'.
  • The sum of these exponents will always equal the power of the entire binomial expression.
  • In our given example, because we expanded \[ (x+y)^{10} \], every resulting term will have exponents on **x** and **y** that add up to 10.
  • Using the term formula, the exponent of x is \[ n-k \], and for y, it鈥檚 \[ k \].
For the 4th term, when the term index \[ k = 3 \], the exponents become \[ x^7 \] and \[ y^3 \], therefore maintaining the balance where the total exponent is still equal to the original power, 10. Understanding how exponents apply helps ensure accurate calculation and expansion of terms.

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

Which two functions have identical graphs, and why? Use a graphing utility to graph the functions in the given order and in the same viewing window. Compare the graphs. (a) \(f(x)=(1-x)^{3}\) (b) \(g(x)=1-x^{3}\) (c) \(h(x)=1+3 x+3 x^{2}+x^{3}\) (d) \(k(x)=1-3 x+3 x^{2}-x^{3}\) (e) \(p(x)=1+3 x-3 x^{2}+x^{3}\)

Use the Binomial Theorem to expand the complex number. Simplify your result. $$(2-3 i)^{6}$$

Prove the identity. \(_{n} P_{n-1}=_{n} P_{n}\)

Three points that are not collinear determine three lines. How many lines are determined by nine points, no three of which are collinear?

Prove the identity. $$_{n} C_{r}=\frac{_{n} P_{r}}{r !}$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.