/*! 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 26 Use long division to divide. \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 26

Use long division to divide. \(\frac{2 x^{3}-4 x^{2}-15 x+5}{(x-1)^{2}}\)

Short Answer

Expert verified
The result of the division is \(2x - 1 - 9 - \frac{15}{x^2 -2x + 1}\).

Step by step solution

01

Polynomial Long Division Setup

To set up the long division, we write \(2x^3 - 4x^2 - 15x + 5\) inside the division symbol and \((x-1)^2 = x^2 - 2x + 1\) outside, such as \(\frac{2x^3 - 4x^2 - 15x + 5}{x^2 - 2x + 1}\).
02

Divide

Now we divide \(2x^3\) by \(x^2\) to get \(2x\). We then multiply \(x^2 - 2x + 1\) by \(2x\) and write it beneath the relevant terms of \(2x^3 - 4x^2 - 15x\), then subtract this from those top terms. We should be careful about the signs here: the subtraction involves changing the signs of all terms. This results in an intermediate polynomial of \(-x^2 - 11x + 5\).
03

Continue Dividing

We repeat what we did in step 2. Divide the first term of the intermediate polynomial (-x^2) by the first term of the divisor (x^2) to get -1. Multiplying \(-1\) by the polynomial \('x^2 -2x + 1'\) gives us \('-x^2 + 2x -1'\). We write this beneath -x^2 - 11x + 5 and subtract, leading to \('-9x + 6'\).
04

Repeat Process

-9 divided by 1 gives -9, hence -9x is now our new divisor for the remaining term. Multiplicating and subtracting gives a leftover of -15.
05

Write Final Answer

The result of the division is then the sum of the terms found at each step, which in this case is \(2x - 1 - 9 - 15\). But since we can't go any further with the division, -15 must remain as the remainder of our division: \(2x - 1 -9 - \frac{15}{x^2 -2x + 1}\).

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!

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

Solve the inequality and graph the solution on the real number line. \(\frac{5}{x-6}>\frac{3}{x+2}\)

Use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\), where \(s\) represents the height of an object (in feet), \(v_{0}\) represents the initial velocity of the object (in feet per second), \(s_{0}\) represents the initial height of the object (in feet), and \(t\) represents the time (in seconds). A projectile is fired straight upward from ground level \(\left(s_{0}=0\right)\) with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet?

The mean salaries \(S\) (in thousands of dollars) of classroom teachers in the United States from 2000 through 2007 are shown in the table. $$\begin{array}{|c|c|}\hline \text { Year } & \text { Salary, } S \\\\\hline 2000 & 42.2 \\ 2001 & 43.7 \\\2002 & 43.8 \\ 2003 & 45.0 \\\2004 & 45.6 \\\2005 & 45.9 \\\2006 & 48.2 \\\2007 & 49.3 \\\\\hline\end{array}$$ A model that approximates these data is given by \(S=\frac{42.6-1.95 t}{1-0.06 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 2000\. (Source: Educational Research Service, Arlington, VA) (a) Use a graphing utility to create a scatter plot of the data. Then graph the model in the same viewing window. (b) How well does the model fit the data? Explain. (c) According to the model, in what year will the salary for classroom teachers exceed \(\$ 60,000 ?\) (d) Is the model valid for long-term predictions of classroom teacher salaries? Explain.

Use a graphing utility to graph the rational function. Give the domain of the function and identify any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. \(h(x)=\frac{12-2 x-x^{2}}{2(4+x)}\)

Determine whether each value of \(x\) is a solution of the inequality. Inequality. \(x^{2}-x-12 \geq 0 \quad\) (a) \(x=5 \quad\) (b) \(x=0\) (c) \(x=-4\) (d) \(x=-3\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

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.