/*! 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 11 Use the binomial theorem to expa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 11

Use the binomial theorem to expand \(\left(3 x^{4}-2 y^{3}\right)^{5}\) into a sum of terms of the form \(c x^{a} y^{b},\) where \(c\) is a real number and \(a\) and \(b\) are nonnegative integers.

Short Answer

Expert verified
The expanded form is \(243x^{20} - 810x^{16}y^3 + 1080x^{12}y^6 - 720x^8y^9 + 240x^4y^{12} - 32y^{15}\).

Step by step solution

01

Understand the Binomial Theorem

The Binomial Theorem states that \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \] where \( \binom{n}{k} \) is a binomial coefficient.
02

Identify the terms

For the given expression \( \left(3 x^{4} - 2 y^{3}\right)^{5} \), identify \( a = 3x^4 \), \( b = -2y^3 \), and \( n = 5 \).
03

Apply the Binomial Theorem

Expand the expression using the Binomial Theorem: \[ \left(3 x^{4} - 2 y^{3}\right)^{5} = \sum_{k=0}^{5} \binom{5}{k} (3 x^{4})^{5-k} (-2 y^{3})^{k} \]
04

Calculate each term

Expand each term separately:For \( k = 0 \): \[ \binom{5}{0} (3x^4)^5 (-2y^3)^0 = 1 \times 243x^{20} \times 1 = 243x^{20} \]For \( k = 1 \): \[ \binom{5}{1} (3x^4)^4 (-2y^3)^1 = 5 \times 81x^{16} \times -2y^3 = -810x^{16}y^3 \]For \( k = 2 \): \[ \binom{5}{2} (3x^4)^3 (-2y^3)^2 = 10 \times 27x^{12} \times 4y^6 = 1080x^{12}y^6 \]For \( k = 3 \): \[ \binom{5}{3} (3x^4)^2 (-2y^3)^3 = 10 \times 9x^8 \times -8y^9 = -720x^8y^9 \]For \( k = 4 \): \[ \binom{5}{4} (3x^4)^1 (-2y^3)^4 = 5 \times 3x^4 \times 16y^{12} = 240x^4y^{12} \]For \( k = 5 \): \[ \binom{5}{5} (3x^4)^{0} (-2y^3)^5 = 1 \times 1 \times -32y^{15} = -32y^{15} \]
05

Write the final expression

Combine all the terms to form the expanded polynomial:\[ \left(3 x^{4} - 2 y^{3} \right)^{5} = 243x^{20} - 810x^{16}y^3 + 1080x^{12}y^6 - 720x^8y^9 + 240x^4y^{12} - 32y^{15} \]

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.

Binomial Theorem
The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions of the form \( (a + b)^n \). This theorem states that any binomial raised to a power can be expanded into a sum involving terms of the form \( \binom{n}{k} a^{n-k} b^{k} \). Here, \( \binom{n}{k} \) are called binomial coefficients, and they play a crucial role in the expansion.

Understanding the Binomial Theorem can simplify otherwise complex polynomial expansions. It is particularly useful in various fields like algebra, combinatorics, and discrete mathematics. Let鈥檚 explore this with an example and a detailed solution.
Binomial Coefficients
In the Binomial Theorem, binomial coefficients \( \binom{n}{k} \) are the numbers that appear in the expansion. These coefficients are represented using combinatorial notation and can be calculated with the formula \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \( n! \) denotes the factorial of \( n \).

For example, in the exercise \( \left( 3x^{4} - 2y^{3} \right)^{5} \), the binomial coefficients are calculated for each term in the expansion. Halfway through solving the problem, we found binomial coefficients 1, 5, 10, and so forth for various \( k \) values. These coefficients help in determining how the expression \( (a + b)^n \) distributes its terms.
Polynomial Expansion
Polynomial expansion is the process of expressing a polynomial as the sum of its terms. The Binomial Theorem provides a systematic way to expand polynomials like \( (3x^4 - 2y^3)^5 \).

In this exercise, we expanded \( \left( 3x^{4} - 2y^{3} \right)^{5} \) using the Binomial Theorem. Let鈥檚 break it down:
  • Identify the terms \( a = 3x^4 \) and \( b = -2y^3 \)
  • Expanded using \( \sum_{k=0}^{5} \binom{5}{k} (3x^4)^{5-k} (-2y^3)^{k} \)
    • For each value of \( k \), we calculate the terms like \( 243x^{20}, -810x^{16}y^3 \), etc., and add them together to form the polynomial expansion.
Discrete Mathematics
Discrete Mathematics studies countable, distinct elements. It includes topics like combinatorics, graph theory, and number theory, among others. The use of binomial coefficients in the Binomial Theorem heavily relies on combinatorial principles, which are an essential part of Discrete Mathematics.

Combinatorics, for example, deals with counting, arrangement, and combination. Calculating \( \binom{n}{k} \) involves understanding these counting principles, useful in various combinatorial problems. By mastering these concepts, students can solve diverse problems in Discrete Mathematics, like expanding binomials or arranging distinct items.

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

How many strings of 20-decimal digits are there that contain two 0s, four 1s, three 2s, one 3, two 4s, three 5s, two 7s, and three 9s?

This procedure is used to break ties in games in the championship round of the World Cup soccer tournament. Each team selects five players in a prescribed order. Each of these players takes a penalty kick, with a player from the first team followed by a player from the second team and so on, following the order of players specified. If the score is still tied at the end of the 10 penalty kicks, this procedure is repeated. If the score is still tied after 20 penalty kicks, a sudden-death shootout occurs, with the first team scoring an unanswered goal victorious. a) How many different scoring scenarios are possible if the game is settled in the first round of 10 penalty kicks, where the round ends once it is impossible for a team to equal the number of goals scored by the other team? b) How many different scoring scenarios for the first and second groups of penalty kicks are possible if the game is settled in the second round of 10 penalty kicks? c) How many scoring scenarios are possible for the full set of penalty kicks if the game is settled with no more than 10 total additional kicks after the two rounds of five kicks for each team?

How many license plates consisting of three letters followed by three digits contain no letter or digit twice?

Suppose that a weapons inspector must inspect each of five different sites twice, visiting one site per day. The inspector is free to select the order in which to visit these sites, but cannot visit site X, the most suspicious site, on two consecutive days. In how many different orders can the inspector visit these sites?

How many ways are there to select 12 countries in the United Nations to serve on a council if 3 are selected from a block of \(45,4\) are selected from a block of \(57,\) and the others are selected from the remaining 69 countries?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics 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.