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Chapter 11: Problem 30

Express each polynomial function in the form \(f(x)=(x-k) q(x)+r\) for the given value of k. $$ f(x)=2 x^{4}+x^{3}-15 x^{2}+3 x ; \quad k=-3 $$

Short Answer

Expert verified
The polynomial is expressed as \((x + 3)(2x^3 - 5x^2 + 3) - 9\).

Step by step solution

01

- Define the Polynomial Division

Identify the polynomial function and the given value of k. In this case, the polynomial is \( f(x) = 2x^4 + x^3 - 15x^2 + 3x \) and \( k = -3 \). We will use polynomial long division to divide \( f(x) \) by \( x + 3 \) (since \( k = -3 \)).
02

- Set Up the Division

Write the polynomial division setup: \( (2x^4 + x^3 - 15x^2 + 3x) \div (x + 3) \). Begin by comparing the leading terms.
03

- Divide the Leading Term

To start, divide the leading term of the polynomial \(2x^4\) by the leading term of \(x + 3\), which is \(x\). This gives \(2x^3\).
04

- Multiply and Subtract

Multiply \(2x^3\) by \(x + 3\) and subtract the result from the original polynomial: \((2x^4 + x^3 - 15x^2 + 3x) - (2x^4 + 6x^3)= -5x^3 - 15x^2 + 3x\)
05

- Repeat the Process

Divide the new leading term \(-5x^3\) by \(x\) to get \(-5x^2\). Multiply and subtract again: \((-5x^3 - 15x^2 + 3x) - (-5x^3 - 15x^2) = 3x\)
06

- Continue

Continue the division process: Divide the new leading term \(3x\) by \(x\) to get \(3\). Multiply ( \(3\)) times (\( x + 3\)) and subtract: \((3x + 0) - (3x + 9) = -9\).
07

- Identify the Quotient and Remainder

The quotient is \(2x^3 - 5x^2 + 3\), and the remainder is \(-9\). Thus, we can write the polynomial in the form: \(f(x) = (x + 3)(2x^3 - 5x^2 + 3) - 9\)

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Key Concepts

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

Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. For example, the given polynomial function in the exercise is:
  • \( f(x) = 2x^4 + x^3 - 15x^2 + 3x \)
Each term in this polynomial function consists of a variable raised to a power and multiplied by a coefficient. Here, the highest power is 4, so this is a fourth-degree polynomial. Polynomial functions can have various degrees based on the highest exponent present.

Specifically:
  • The 'degree' of a polynomial is the highest power of the variable.
  • The 'leading coefficient' is the coefficient of this highest power.
In the example, the leading coefficient is 2. Understanding these basics makes polynomial division more manageable.
Remainder Theorem
The Remainder Theorem links polynomial division to evaluating polynomial functions. It states that for any polynomial \( f(x)\) divided by \( x - k \), the remainder of this division is equal to \( f(k) \).
Applying this to our exercise, if we want to find the remainder when \( f(x) = 2x^4 + x^3 - 15x^2 + 3x \) is divided by \( x + 3 \) (where \(k = -3\)), we evaluate the polynomial at \(k = -3\).
Here's how:
  • Calculate \( f(-3) \):
  • \( f(-3) = 2(-3)^4 + (-3)^3 - 15(-3)^2 + 3(-3) \)
  • Simplify step by step:
  • \( = 2(81) + (-27) - 15(9) - 9 = 162 - 27 - 135 - 9 \)
  • \( = -9 \)
The remainder is, therefore, \(-9\), which aligns with the earlier division result. This method saves time in polynomial division checking.
Dividing Polynomials
Dividing polynomials involves breaking down a complex polynomial by another, simpler polynomial known as the divisor. There are multiple methods to achieve this, but polynomial long division is a common approach.
Here's a quick summary of how we expressed the polynomial \( f(x) = 2x^4 + x^3 - 15x^2 + 3x \) in the form \( f(x) = (x + 3)q(x) + r \) where \( k = -3 \):
  • First, set up the division of \( (2x^4 + x^3 - 15x^2 + 3x) \) by \( (x + 3) \).
  • Compare leading terms and divide: \( 2x^4 / x = 2x^3 \).
  • Multiply and subtract: \( (2x^4 + x^3 - 15x^2 + 3x) - (2x^4 + 6x^3) = -5x^3 - 15x^2 + 3x \).
  • Repeat the process with new leading terms: \( -5x^3 / x = -5x^2 \), and so on, until reaching the end.
Eventually, we find a quotient polynomial \( q(x) = 2x^3 - 5x^2 + 3 \) and a remainder \( r = -9 \).
Thus, the polynomial function can be rewritten as:
  • \( f(x) = (x + 3)(2x^3 - 5x^2 + 3) - 9 \)

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Most popular questions from this chapter

Approximate all real zeros of each function to the nearest hundredth. \(f(x)=-\sqrt{15} x^{4}-\sqrt{3} x^{2}+7\)

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 \(55-64 ?\)

Approximate to the nearest hundredth the coordinates of the turning point in the given interval of the graph of each polynomial function. \(f(x)=x^{3}+4 x^{2}-8 x-8, \quad[-3.8,-3]\)

The table shows the worldwide number of Amazon.com employees (in thousands) from 2009 to 2017. $$ \begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Amazon.com employees } \\ \text { (in thousands) } \end{array} \\ 2009 & 24.3 \\ 2010 & 33.7 \\ 2011 & 56.2 \\ 2012 & 88.4 \\ 2013 & 117.3 \\ 2014 & 154.1 \\ 2015 & 230.8 \\ 2016 & 341.4 \\ 2017 & 541.9 \\ \hline \end{array} $$ (a) Using \(x=0\) to represent \(2009, x=1\) to represent \(2010,\) and so on, use the regression feature of a calculator to determine the quadratic function that best fits the data. Give coefficients to the nearest hundredth. (b) Repeat part (a) for a cubic function (degree 3). Give coefficients to the nearest hundredth. (c) Repeat part (a) for a quartic function (degree 4 ). Give coefficients to the nearest hundredth. (d) Compare the correlation coefficient \(R^{2}\) for the three functions in parts (a)-(c) to determine which function best fits the data. Give its value to the nearest ten-thousandth.

Which choice has a graph that does not have a horizontal asymptote? A. \(f(x)=\frac{2 x-7}{x+3}\) B. \(f(x)=\frac{3 x}{x^{2}-9}\) C. \(f(x)=\frac{x^{2}-9}{x+3}\) D. \(f(x)=\frac{x+5}{(x+2)(x-3)}\)
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