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Chapter 2: Problem 44

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

Short Answer

Expert verified
The result of the synthetic division is the polynomial \( -x^2 - x + 3 \) with remainder 2.

Step by step solution

01

Write Down Coefficients

Write down the coefficients of the cubic polynomial in descending order of powers, and the constant term from the linear factor. For the polynomial \( -x^3+2x^2-3x+5 \), the coefficients are [-1, 2, -3, 5] and from \( x+1 \), the constant is -1.
02

Setup for Synthetic Division

Set up the coefficients of the polynomial in a horizontal line, and write the constant term (negative) from the divisor on the left side. The setup should look like this: \[ \begin{array}{|c| c c c c|}\multicolumn{1}{r}{-1} & -1 & 2 & -3 & 5 \\end{array} \]
03

Perform Synthetic Division

Now, perform synthetic division: bring down the first term straight, multiply the divisor's constant with it and write the result under the next term. Add up vertically. Repeat the process until you reach the end. \[ \begin{array}{c| c c c c} -1 & -1 & 2 & -3 & 5 \ & 1 & -3 & 6 & -3\ \hline & -1 & -1 & 3 & 2 \ \end{array} \]
04

Interpret the Result

The bottom row of numbers would be the coefficients of the resulting polynomial after division. Starting from one degree lower than the original polynomial, the quotient is \(-x^2 -x +3\). The final number '2' is the remainder and completes the equation as follows: \(-x^3+2x^2-3x+5 = (-x^2 -x +3)(x+1) +2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division
Polynomial division is akin to long division that you might remember from grade school, but instead of dividing numbers, we are dealing with polynomials. In this process, you divide one polynomial by another鈥攗sually of a lower degree鈥攁nd find either a quotient and a remainder or sometimes a quotient that divides evenly with no leftover.

Synthetic division is a shortcut method specifically used when dividing polynomials by linear factors of the form \( x - c \). This method is generally faster and simpler than traditional polynomial division methods. It skips steps and reduces complexity by focusing only on the coefficients.
  • Polynomials must follow a specific order, with descending powers of variables, for accurate division.
  • Results can verify factors or roots of the original polynomial.
Coefficients
Coefficients are vital numbers that multiply the variable terms in a polynomial, defining the magnitude and direction of each term. In the polynomial \( -x^3 + 2x^2 - 3x + 5 \), the coefficients are -1, 2, -3, and 5.

When using synthetic division, you list these coefficients out in a row to commence the process, while using the constant from the divisor. An essential step is lining them up correctly, as any tiny mistake can lead to errors in division.
  • Check polynomial terms are in the correct descending order, otherwise, you may miss some coefficients during division.
  • Remain attentive to the signs (+/-) as these directly affect the result.
Remainder Theorem
The Remainder Theorem is a fundamental concept that connects polynomial division and evaluating polynomials. It states that when you divide a polynomial \( f(x) \) by a linear divisor of the form \( x - c \), the remainder of this division is precisely \( f(c) \). In practical terms, the result gives you quick insights without full computation.

In the performed division, the remainder 2 tells us that when the polynomial \( -x^3 + 2x^2 - 3x + 5 \) is evaluated at \(-1\), the output is 2. This quick check confirms the computations and the quotient accuracy, giving confidence in the validity of results.
  • The remainder is a crucial part of polynomial division results, indicating if we have division with "no remainder" or how much "leftover" remains.
  • Provides an immediate check for potential divisors or roots.
Polynomial Long Division
Polynomial long division is like the written long division method you learned in elementary school. Though synthetic division is preferred for its simplicity, knowing polynomial long division is beneficial for more complex divisions.

This method involves a step-by-step division process, dealing explicitly with each term of the dividend and divisor, adjusting to subtract and bring down terms just like in numerical long division.
  • Long division is detailed and systematic, covering each polynomial term.
  • Even though it is tedious, it is flexible and works for any degree or form polynomial divisor.
Understanding both synthetic division and long-form division equips you with versatile skills necessary for tackling a wide range of polynomial problems.

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Most popular questions from this chapter

Home Mortgage The total interest \(u\) paid on a home mortgage of \(P\) dollars at interest rate \(r\) for \(t\) years is $$u=P\left[\frac{r t}{1-\left(\frac{1}{1+r / 12}\right)^{12 t}}-1\right]$$ Consider a \(\$ 120,000\) home mortgage at 7\(\frac{1}{2} \%\) (a) Use a graphing utility to graph the total interest function. (b) Approximate the length of the mortgage for which the total interest paid is the same as the size of the mortgage. Is it possible that some people are paying twice as much in interest charges as the size of the mortgage?

Cost The ordering and transportation cost \(C\) (in thousands of dollars) for machine parts is $$C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right), \quad x \geq 1$$ where \(x\) is the order size (in hundreds). In calculus, it can be shown that the cost is a minimum when $$3 x^{3}-40 x^{2}-2400 x-36,000=0$$ Use a calculator to approximate the optimal order size to the nearest hundred units.

Modeling Polynomials Sketch the graph of a polynomial function that is of fifth degree, has a zero of multiplicity \(2,\) and has a negative leading coefficient. Sketch the graph of another polynomial function with the same characteristics except that the leading coefficient is positive.

Intensity of Sound In Exercises \(47-50\) , use the following information for determining sound intensity. The level of sound \(\beta,\) in decibels, with an intensity of \(I,\) is given by \(\beta=10 \log \left(I I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 47 and \(48,\) find the level of sound \(\beta .\) Due to the installation of a muffler, the noise level of an engine decreased from 88 to 72 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of the muffler.

Intensity of Sound In Exercises \(47-50\) , use the following information for determining sound intensity. The level of sound \(\beta,\) in decibels, with an intensity of \(I,\) is given by \(\beta=10 \log \left(I I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 47 and \(48,\) find the level of sound \(\beta .\) Due to the installation of noise suppression materials, the noise level in an auditorium decreased from 93 to 80 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of these materials.
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