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Chapter 11: Problem 32

For each polynomial function, use the remainder theorem and synthetic division to find \(f(k) .\) $$ f(x)=x^{2}+5 x+6 ; \quad k=-2 $$

Short Answer

Expert verified
\( f(-2) = 0 \).

Step by step solution

01

- Set Up the Synthetic Division

Set up synthetic division for the polynomial function. Arrange the coefficients of each term of the polynomial: The coefficients of the polynomial function \[f(x) = x^2 + 5x + 6\] are 1 (for \(x^2\)), 5 (for \(x\)), and 6 (constant term). The value of \(k\) provided is -2.
02

- Perform the Synthetic Division

Write -2 on the left side and the coefficients (1, 5, 6) on the right side. Then, perform synthetic division:1. Bring down the 1.2. Multiply -2 by 1 to get -2.3. Add 5 and -2 to get 3.4. Multiply -2 by 3 to get -6.5. Add 6 and -6 to get 0.The operations will look like this:\[\begin{array}{r|rrr}-2 & 1 & 5 & 6 \hline & 1 & 3 & 0 \end{array}\]
03

- Interpret the Remainder

The remainder is the final value obtained after performing synthetic division. In this case, it is 0. According to the remainder theorem, this remainder is equal to \( f(k) \).
04

- Conclude with the Result

Since the remainder is 0, using the remainder theorem, \( f(-2) = 0 \).

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Key Concepts

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

Polynomial Functions
A polynomial function is simply a function that is made up of one or more terms, where each term includes a variable raised to a non-negative integer power. For example, the function \(f(x) = x^2 + 5x + 6\) is a polynomial function. Here's how it breaks down:
  • The highest power of the variable (x) is 2, which tells us it's a quadratic polynomial.
  • Each term can be separated: \(x^2 + 5x + 6\).
  • The coefficients are the numbers in front of the variables: 1 for \(x^2\), 5 for \(x\), and 6 for the constant term.
Polynomial functions are widely used in Algebra to model various real-world scenarios and are essential for understanding the behavior of different functions.
Synthetic Division
Synthetic division is a simplified way of dividing a polynomial by a binomial of the form \(x - k\). It's faster and simpler compared to traditional long division. Let's discuss the process step-by-step.

Step 1: Setup
Write down the coefficients of the polynomial. For \(f(x) = x^2 + 5x + 6\), the coefficients are 1, 5, and 6.
Place the value of \(k\) (which is -2 in the example) to the left side.

Step 2: Perform Synthetic Division
  • Bring down the first coefficient (1).
  • Multiply it by \(k\) (which is -2) to get -2. Add this to the next coefficient (5) to get 3.
  • Repeat the process with the result: multiply 3 by -2 to get -6, then add this to the final coefficient (6) to get 0.
The final value on the bottom row is the remainder, which in this case is 0.

Synthetic division provides a quick method to simplify polynomial division and is immensely useful in Algebra.
Algebra
Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. Its basics include working with variables, arithmetic operations, and understanding polynomial functions.

Key Concepts in Algebra:
  • Variables: These are symbols used to represent unknown values, generally letters like x or y.
  • Coefficients: Numbers multiplying the variables in a polynomial (like 5 in \(5x\)).
  • Constants: Fixed values that don't change, like 6 in \(x^2 + 5x + 6\).
Algebra helps us solve equations and understand how variables interact with each other.

Understanding Algebra is crucial since it forms the foundation for advanced math topics and real-world applications, from calculating interest rates to coding algorithms. By mastering synthetic division and functions, you enhance your ability to handle more complicated algebraic expressions and equations.

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Most popular questions from this chapter

The table shows the total (cumulative) number of ebola cases reported in Sierra Leone during a serious West African ebola outbreak in \(2014-2015 .\) The total number of cases is reported \(x\) months after the start of the outbreak in May \(2014 .\) $$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Months after } \\ \text { May 2014 } \end{array} & \text { Total Ebola Cases } \\ \hline 0 & 16 \\ 2 & 533 \\ 4 & 2021 \\ 6 & 7109 \\ 8 & 10,518 \\ 10 & 11,841 \\ 12 & 12,706 \\ 14 & 13,290 \\ 16 & 13,823 \\ 18 & 14,122 \\ \hline \end{array} $$ (a) Use the regression feature of a calculator to determine the quadratic function that best fits the data. Let \(x\) represent the number of months after May \(2014,\) and let \(y\) represent the total number of ebola cases. 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.

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

Use synthetic division to divide. $$ \frac{x^{5}+x^{4}+x^{3}+x^{2}+x+3}{x+1} $$

Use a graphing calculator to find (or approximate) the real zeros of each function \(f(x)\). Express decimal approximations to the nearest hundredth. \(f(x)=\sqrt{10} x^{3}-\sqrt{11} x-\sqrt{8}\)

Use synthetic division to divide. $$ \frac{4 x^{2}-5 x-20}{x-4} $$
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