/*! 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 36 For the following exercises, use... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 36

For the following exercises, use synthetic division to find the quotient. $$ \left(4 x^{4}-2 x^{3}-4 x+2\right) \div(2 x-1) $$

Short Answer

Expert verified
The quotient is \(4x^3 - 4\) with a remainder of 0.

Step by step solution

01

Set Up the Synthetic Division

To perform synthetic division, identify the root of the divisor \((2x-1)\), which is \(x = \frac{1}{2}\). Write down the coefficients of the dividend \(4x^4 - 2x^3 + 0x^2 - 4x + 2\). Note that we include a 0 for the missing \(x^2\) term. The coefficients are \([4, -2, 0, -4, 2]\).
02

Bring Down the Leading Coefficient

Start the synthetic division by bringing down the leading coefficient, which is 4, to the bottom row.
03

Multiply and Add

Multiply the root \(\frac{1}{2}\) by the leading coefficient 4 and place it under the next coefficient (-2). Add -2 and 2 to get 0. Continue the process by multiplying \(\frac{1}{2}\) by the result (0) and add to the next coefficient (0). Repeat this for all coefficients. You should have:- Multiply \(\frac{1}{2}\) by 4, add to -2, to get 0.- Multiply \(\frac{1}{2}\) by 0, add to 0, to get 0.- Multiply \(\frac{1}{2}\) by 0, add to -4, to get -4.- Multiply \(\frac{1}{2}\) by -4, add to 2, to get 0.
04

Interpret the Result

The numbers in the bottom row represent the coefficients of the quotient polynomial: \(4x^3 + 0x^2 + 0x - 4\). The last number (0) is the remainder.

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.

Polynomials
Polynomials are mathematical expressions consisting of variables and coefficients, structured in a way that they include terms of the form \( ax^n \).These terms are combined using addition, subtraction, and multiplication operators.Here are some key points to understand polynomials:
  • The variable \( x \) can have non-negative integer exponents.
  • A polynomial is usually expressed in a standard form, ordered from the highest power of \( x \) to the lowest.
  • Each term in a polynomial contains a coefficient, which is a real number.
  • The highest exponent of the variable in the polynomial indicates its degree. For example, a degree four polynomial will have the highest term as \( x^4 \).
In the given exercise, the expression \( 4x^4 - 2x^3 - 4x + 2 \) is a polynomial.It has four terms, and its degree is 4 because the highest power of \( x \) is 4.
Roots of Polynomials
The roots of a polynomial are the solutions where the polynomial equals zero.Finding roots is essential in simplifying and solving polynomial equations.Here are some important aspects about polynomial roots:
  • For a polynomial equation \( P(x) = 0 \), the roots are the values of \( x \) that make the equation true.
  • The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) will have exactly \( n \) roots, which may be real or complex, some of which might be repeated.
  • In synthetic division, verifying roots is essential as the method is used to divide polynomials by linear factors \((x - r)\), where \( r \) is a root of the polynomial.
In the exercise, the divisor is \( 2x-1 \), extracted from which we find a root \( x = \frac{1}{2} \).This root is then used in the synthetic division process.
Synthetic Division Steps
Synthetic division is a handy method for dividing polynomials when the divisor is in the linear form \((x - r)\).The approach simplifies long division by focusing on coefficients.Here's how to perform synthetic division:
  • First, determine the root \( r \) from the divisor, such as \((2x-1)\), which leads to \( r = \frac{1}{2} \).
  • Next, list the coefficients of the polynomial in descending order of power. Don鈥檛 forget to include 0 for any missing terms.
  • Place the root \( \frac{1}{2} \) to the left of the setup. Bring down the first coefficient directly to the bottom row.
  • Multiply this coefficient by the root \( r \) and carry the result to the line of coefficients, then add it to the next coefficient.
  • Continue the process: multiply the new number in the bottom row by \( r \) and add it to the next coefficient above, repeating till the end.
  • The resulting numbers at the bottom row are the coefficients of the quotient and any remainder.
In the solution given, these steps yield the result \( 4x^3 + 0x^2 + 0x - 4 \) for the quotient, and a remainder of 0.

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

For the following exercises, use the given information to answer the questions. The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

For the following exercises, use the given information to answer the questions. The distance \(s\) that an object falls varies directly with the square of the time, \(t,\) of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

For the following exercises, use a calculator to graph the equation implied by the given variation. \(y\) varies directly as the square root of \(x\) and when \(x=36, y=2\).

For the following exercises, find the inverse of the functions with \(a,\) c positive real numbers. $$f(x)=\sqrt[3]{a x+b}$$

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$f(x)=(x-4)^{2}, x \geq 4$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.