/*! 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 18 For the following exercises, use... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 18

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (3 x-2 y)^{4} $$

Short Answer

Expert verified
\((3x - 2y)^4 = 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4\)."

Step by step solution

01

Write out the Binomial Theorem

The Binomial Theorem is written as \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This will help us expand any binomial expression of the form \((a+b)^n\). For our problem, \(a = 3x\), \(b = -2y\), and \(n = 4\).
02

Determine the Binomial Coefficients

Use the formula for binomial coefficients \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) for each term where \(k\) ranges from 0 to 4, so \(n = 4\). The coefficients are \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), and \(\binom{4}{4} = 1\).
03

Expand Each Term of the Binomial

Using the binomial theorem, calculate each term for the binomial expansion:- For \(k = 0\): \(1 \cdot (3x)^4 \cdot (-2y)^0 = 81x^4\)- For \(k = 1\): \(4 \cdot (3x)^3 \cdot (-2y)^1 = -216x^3y\)- For \(k = 2\): \(6 \cdot (3x)^2 \cdot (-2y)^2 = 216x^2y^2\)- For \(k = 3\): \(4 \cdot (3x)^1 \cdot (-2y)^3 = -96xy^3\)- For \(k = 4\): \(1 \cdot (3x)^0 \cdot (-2y)^4 = 16y^4\)
04

Write the Complete Expansion

Combine all the terms from Step 3 to get the full binomial expansion: \((3x - 2y)^4 = 81x^4 - 216x^3y + 216x^2y^2 - 96xy^3 + 16y^4\).

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 Expansion
The binomial expansion is a way to express the power of a binomial as a sum of terms. If you have an expression like \((a + b)^n\), the Binomial Theorem allows us to expand it into a series of terms. Each term consists of a binomial coefficient, a power of \(a\), and a power of \(b\). It's very useful for simplifying expressions in algebra.

In our exercise, we expanded \((3x - 2y)^4\) using this very method. By doing so, we transformed a complex expression into a more manageable form, making it easier to work with.
Binomial Coefficients
Binomial coefficients are essential in determining each term's weight in a binomial expansion. They tell us how many ways we can choose elements from a set, which correlates to the number of times each product of \(a\) and \(b\) appears in the expansion.

These coefficients are calculated using the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

For our specific problem, we calculated the binomial coefficients for values of \(k\) from 0 to 4, arriving at the sequence: 1, 4, 6, 4, and 1. These coefficients correspond to the different terms in the expansion process.
Polynomials
A polynomial is simply an expression with multiple terms, where each variable has non-negative integer exponents. It's fundamental in algebra to handle these mathematical expressions gracefully.

The concept of a polynomial is crucial for understanding binomial expansion because after expanding a \((a + b)^n\), the resulting expression is essentially a polynomial.

In our exercise, once we expanded the binomial, we ended up with a polynomial of five terms: 81x^4, -216x^3y, 216x^2y^2, -96xy^3, and 16y^4.
Algebra
Algebra is a wide and foundational branch of mathematics focused on symbols and the rules for manipulating these symbols. It provides a clear and structured way to represent numbers and their relationships.
  • Symbols such as \(x\) and \(y\) allow for the generalization of arithmetic.
  • Through algebra, we can solve equations, expand expressions, and explore a wide range of mathematical principles.
In our exercise, we used algebraic principles to expand \((3x - 2y)^4\) into a series of terms, making it simpler to analyze or compute when needed. Algebra helps bridge the gap between pure arithmetic numbers and more abstract mathematical concepts.

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

The sum of an infin te geometric series is five times the value of the fi st term. What is the common ratio of the series?

Use this data for the exercises that follow: In 2013 , there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{[34]}\) If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)

For the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: A heart or a non-jack

For the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: A two

A conductor needs 5 cellists and 5 violinists to play at a diplomatic event. To do this, he ranks the orchestra's 10 cellists and 16 violinists in order of musical proficie cy. What is the ratio of the total cellist rankings possible to the total violinist rankings possible?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

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.