/*! 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 Express as a polynomial. $$(x-... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 11

Express as a polynomial. $$(x-2 y)^{3}$$

Short Answer

Expert verified
The polynomial is \(x^3 - 6x^2y + 12xy^2 - 8y^3\).

Step by step solution

01

Understand the Problem

We need to expand the expression \((x - 2y)^3\) into a polynomial. This requires us to apply the binomial theorem or method of expansion for a cubic binomial.
02

Apply the Binomial Theorem

The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = x\), \(b = -2y\), and \(n = 3\).
03

Calculate the Binomial Coefficients

For \(n = 3\), the coefficients are given by the binomial coefficients: \(\binom{3}{0} = 1\), \(\binom{3}{1} = 3\), \(\binom{3}{2} = 3\), \(\binom{3}{3} = 1\).
04

Write the Binomial Expansion

Using the coefficients and expanding the terms, the expression becomes:- \(\binom{3}{0} x^3 (-2y)^0 = x^3\)- \(\binom{3}{1} x^2 (-2y)^1 = 3x^2(-2y) = -6x^2y\)- \(\binom{3}{2} x^1 (-2y)^2 = 3x(4y^2) = 12xy^2\)- \(\binom{3}{3} x^0 (-2y)^3 = (-8y^3)\)
05

Combine All Terms

Combine all terms from the expansion:\(x^3 - 6x^2y + 12xy^2 - 8y^3\). This is the polynomial form of \((x - 2y)^3\).

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.

Binomial Theorem
The binomial theorem is a fundamental principle used to expand expressions that are raised to a power. This theorem provides a formula for expanding expressions of the form \((a + b)^n\). It can be expressed as:
  • \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
Here, \(a\) and \(b\) are any numbers or variables, and \(n\) is a non-negative integer.

Each term in this expansion consists of:
  • A binomial coefficient, \(\binom{n}{k}\), which helps determine the specific weight of the terms.
  • The term \(a^{n-k}\), indicating how many times \(a\) is multiplied.
  • The term \(b^k\), showing how many times \(b\) is multiplied.
Using this theorem simplifies expanding powers of binomials into a sum of more manageable terms.
Cubic Binomials
A cubic binomial is a specific case where the binomial theorem is applied to an expression raised to the third power, such as \((x - 2y)^3\). This results in four distinct terms, each having a different degree. Here鈥檚 a breakdown of expanding a cubic binomial:
  • Identify the terms \(a\) and \(b\). In this case, \(a = x\) and \(b = -2y\).
  • Recognize \(n = 3\) as the power to which the binomial is raised.
  • Use the binomial theorem to expand and compute each term using previously calculated binomial coefficients.
The expansion results in a polynomial that can be expressed through separate terms, each derived by specific calculations based on the values of \(a\), \(b\), and the coefficients according to their respective powers.
Binomial Coefficients
Binomial coefficients are a key element in the expansion of binomials. They are denoted as \(\binom{n}{k}\) and can be calculated using the formula:
  • \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
Where \(n!\) is the factorial of \(n\), a product of all positive integers up to \(n\), and \(k\) is a particular term's order in the expansion.

For example, when expanding \((x - 2y)^3\), we use the following coefficients:
  • \(\binom{3}{0} = 1\)
  • \(\binom{3}{1} = 3\)
  • \(\binom{3}{2} = 3\)
  • \(\binom{3}{3} = 1\)
These coefficients serve as multipliers for each term of the expanded binomial, dictating the contributing influence each part of \(a\) and \(b\) has on the outcome. The expansion thus integrates these coefficients to form a polynomial.

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

Rationalize the numerator. $$\frac{\sqrt{b}+\sqrt{c}}{b^{2}-c^{2}}$$

Simplify the expression. $$\frac{\frac{1}{x+h}-\frac{1}{x}}{h}$$

O'Carroll's formula is used to handicap weight lifters. If a lifter who weighs \(b\) kilograms lifts \(w\) kilograms of welght, then the handicapped weight \(W\) is given by $$W=\frac{w}{\sqrt{b-35}}$$ Suppose two lifters weighing 75 kilograms and 120 kilograms lift weights of 180 kilograms and 250 kilograms, respectively. Use O'Carroll's formula to determine the superior weight lifter.

Simplify the expression. $$=(3 x+2)^{1 / 3}(2)(4 x-5)(4)+(4 x-5)^{2}\left(\frac{1}{3}\right)(3 x+2)^{-2 / 3}(3)$$

Simplify the expression, assuming \(x\) and \(y\) may be negative. b$$\sqrt[4]{(x+2)^{12} y^{4}}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.