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Chapter 3: Problem 34

For each polynomial function, use the remainder theorem and synthetic division to find \(f(k) .\) See Example 2 $$f(x)=x^{4}+6 x^{3}+9 x^{2}+3 x-3 ; k=4$$

Short Answer

Expert verified
f(4) = 801

Step by step solution

01

- Set up Synthetic Division

To use synthetic division, write down the coefficients of the polynomial. For the polynomial function \( f(x) = x^4 + 6x^3 + 9x^2 + 3x - 3 \), the coefficients are: [1, 6, 9, 3, -3].
02

- Write the Value of k

Write the value of \( k \) to the left. In this case, \( k = 4 \).
03

- Begin Synthetic Division

Bring down the first coefficient (1).
04

- Multiply and Add

Multiply \( 4 \) (the value of \( k \)) by the first coefficient brought down (1), and write the answer under the second coefficient (6). Add this product to the second coefficient. Repeat this process for each coefficient.\[\begin{array}{r|rrrrr}4 & 1 & 6 & 9 & 3 & -3 \hline & & 4 & 40 & 196 & 804 \& 1 & 10 & 49 & 196 & 801 \end{array}\]
05

- Interpret the Result

The last number in the bottom row of the synthetic division process is the remainder. This remainder is the value of \( f(4) \).

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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 expressions consisting of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents of variables.
In general, a polynomial function can be written as:
\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_2 x^2 + a_1 x + a_0 \]. Here,
  • \(a_n, a_{n-1}, ..., a_0\) are the coefficients of the polynomial.
  • \(x\) is the variable.
  • \(n\) is the degree of the polynomial, the highest power of the variable in the polynomial.

For example, in the given polynomial \( f(x) = x^4 + 6x^3 + 9x^2 + 3x - 3 \), the degree is 4, and the coefficients are 1, 6, 9, 3, and -3 respectively.
Remainder theorem
The remainder theorem is a useful concept in algebra, especially when dealing with polynomials.
It states that if a polynomial \( f(x) \) is divided by \( x - k \), the remainder of this division is equal to \( f(k) \).
This means that instead of performing a full polynomial division, you can simply substitute the value of \( k \) into the polynomial to find the remainder.
In our example, using the polynomial \( f(x) = x^4 + 6x^3 + 9x^2 + 3x - 3 \) and \( k = 4 \), the remainder theorem tells us that the remainder when \( f(x) \) is divided by \( x - 4 \) is equal to \( f(4) \).
This makes evaluating polynomials at specific values much easier and quicker.
Evaluating polynomials
Evaluating polynomials means finding the value of the polynomial function for a specific value of the variable.
One effective method to perform this task is by using synthetic division.
Here's a step-by-step guide for using synthetic division to evaluate a polynomial, like in our example:
  • First, list the coefficients of the polynomial. For \( f(x) = x^4 + 6x^3 + 9x^2 + 3x - 3 \), they are [1, 6, 9, 3, -3].
  • Write the value of \( k \) to the left, which is 4 in this case.
  • Bring down the first coefficient (1). It remains unchanged.
  • Multiply the value of \( k \) by the first coefficient and add it to the next coefficient. Repeat for all coefficients.
  • The result of the final step is the value of \( f(k) \). In our example, \( f(4) = 801 \).

Using synthetic division makes polynomial evaluation simpler, faster, and helps in verifying the remainder theorem quickly.

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Most popular questions from this chapter

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier chapters. Work Exercises \(119-128\) in order. (a) Given that 1 and 2 are zeros of the denominator, factor the denominator completely. (b) Write the entire quotient for \(f\) so that the numerator and the denominator are in factored form.

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier chapters. Work Exercises \(119-128\) in order. (a) What is the common factor in the numerator and the denominator? (b) For what value of \(x\) will there be a point of discontinuity (i.e., a "hole")?

Solve each problem. Sum and Product of Two Numbers Find two numbers whose sum is 20 and whose product is the maximum possible value. (Hint: Let \(x\) be one number. Then \(20-x\) is the other number. Form a quadratic function by multiplying them, and then find the maximum value of the function.

Solve each problem.Current in a Circuit Speed of a Pulley The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in.

Explain how the graph of each function can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}} .\) Then graph \(f\) and give the (a) domain and (b) range. Determine the intervals of the domain for which the function is ( \(c\) ) increasing or (d) decreasing. See Examples \(1-3\). $$f(x)=\frac{1}{x-3}$$
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