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Chapter 11: Problem 1

Choose the letter of the correct setup to perform synthetic division on the indicated quotient. $$ \frac{x^{2}+3 x-6}{x-2} $$ A. \(- 2 )\overline{ 1 3 - 6 }\) B. \(- 2 )\overline { - 1 } - 3 6\) C. \(2 )\overline { 1 1 3 - 6 }\) D. \(2 )\overline { - 1 - 3 6 }\)

Short Answer

Expert verified
Option C

Step by step solution

01

Identify the Divisor and Dividend

The given expression to divide is \(\frac{x^{2} + 3x - 6}{x - 2}\). Here, the divisor is \(x - 2\) and the dividend is \(x^{2} + 3x - 6\).
02

Determine the Divisor's Root

To use synthetic division, the root of the divisor \(x - 2\) is needed. The root is found by setting the divisor equal to zero: \(x - 2 = 0 \Rightarrow x = 2\).
03

Write Down the Coefficients of the Dividend

List the coefficients of the dividend \(x^{2} + 3x - 6\). They are: \[1, 3, -6\].
04

Set Up the Synthetic Division

Using the root \(2\) from the divisor and the coefficients \(1, 3, -6\), set up the synthetic division as follows: \(2 )\begin{array}{|c c c c|} 1 & 3 & -6 \ \ \ \ \end{array}\).
05

Choose the Correct Option

Compare the setup with the given options. The setup matches option C: \(2 )\begin{array}{|c c c c|} 1 & 3 & -6 \ \ \ \ \end{array}\).

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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 a method used to divide one polynomial by another. If you are familiar with long division of numbers, polynomial division is quite similar but with polynomials instead of numbers. In this exercise, we used synthetic division, a simplified form of polynomial division specifically used when the divisor is a linear polynomial of the form \(x - a\). Synthetic division makes the division process quicker and reduces the chance of errors.
To perform synthetic division, follow these steps:
  • Identify the divisor and the dividend.
  • Determine the root of the divisor by setting it to zero.
  • List the coefficients of the dividend.
  • Set up the synthetic division using the root and coefficients.
  • Carry out the synthetic division.
Synthetic division provides a more straightforward and less error-prone solution compared to traditional polynomial long division.
divisor and dividend
In polynomial division, the terms 'divisor' and 'dividend' are crucial. The dividend is the polynomial you are dividing into segments, and the divisor is the polynomial by which you are dividing. In our exercise, the given expression is \( \frac{x^2+3x-6}{x-2} \). Here:
  • The dividend is \(x^2 +3x - 6\).
  • The divisor is \(x - 2\).
Understanding the roles of the divisor and dividend helps in setting up the division process correctly. You first find the root of the divisor by setting \(x - 2 \) to zero, giving you \(x = 2\). This root is then used in the synthetic division setup along with the coefficients of the dividend.
coefficients
Coefficients are the numerical parts of the terms in a polynomial. They are essential in synthetic division as they form the basis of the division setup. For the polynomial \(x^2 + 3x - 6\), the coefficients are:
  • 1 for \(x^2\)
  • 3 for \(x\)
  • -6 for the constant term
Listing these coefficients accurately is critical to successful synthetic division. Here鈥檚 how you do it:
Write down the coefficients in the order of the polynomial鈥檚 degree. For our polynomial \(x^2 + 3x - 6\), list them as \[1, 3, -6\]. Using these coefficients in the synthetic division setup helps streamline the calculation process and ensures that the division is accurate.
roots of polynomial
The root of a polynomial is the value of \(x\) that makes the polynomial equal to zero. In the context of synthetic division, finding the root of the divisor is a key step. For our divisor \(x - 2\), the root is found by solving the equation \(x - 2 = 0\), which gives \(x = 2\). This root is then used to set up the synthetic division.
Polynomials can have multiple roots, and these roots are the values that the polynomial functions cross the x-axis. To recap:
  • Set the divisor equal to zero to find its root.
  • Use this root in the synthetic division setup.
  • Understanding the roots of polynomials helps in various algebraic operations, including factoring and solving polynomial equations.
By grasping the role of roots in synthetic division, you can solve polynomial division problems more effectively.

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Most popular questions from this chapter

Use synthetic division to determine whether the given number is a zero of the polynomial function. $$ 2-i ; \quad f(x)=x^{2}+3 x+4 $$

For each of the following, (a) show that the polynomial function has a zero between the two given integers and (b) approximate all real zeros to the nearest thousandth. \(f(x)=x^{4}-4 x^{3}-20 x^{2}+32 x+12 ;\) between -4 and -3

The table contains incidence ratios by age for deaths due to coronary heart disease (CHD) and lung cancer (LC) when comparing smokers ( \(21-39\) cigarettes per day) to nonsmokers. $$\begin{array}{|c|c|c|}\hline \text { Age } & \text { CHD } & \text { LC } \\\\\hline 55-64 & 1.9 & 10 \\\\\hline 65-74 & 1.7 & 9 \\\\\hline\end{array}$$ The incidence ratio of 10 means that smokers are 10 times more likely than nonsmokers to die of lung cancer between the ages of 55 and \(64 .\) If the incidence ratio is \(x,\) then the percent \(P\) (in decimal form) of deaths caused by smoking can be calculated using the rational function$$P(x)=\frac{x-1}{x}$$ (Data from Walker, A., Observation and Inference: An Introduction to the Methods of Epidemiology, Epidemiology 91影视 Inc.) (a) As \(x\) increases, what value does \(P(x)\) approach? (b) Why might the incidence ratios be slightly less for ages \(65-74\) than for ages \(55-64 ?\)

Show that the real zeros of each polynomial function satisfy the given conditions. \(f(x)=2 x^{5}-x^{4}+2 x^{3}-2 x^{2}+4 x-4 ; \quad\) no real zero greater than 1

Use synthetic division to divide. $$ \frac{x^{3}+2 x^{2}-4 x+3}{x-4} $$
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