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Chapter 3: Problem 29

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

Short Answer

Expert verified
The value of \(f(2)\) is -1.

Step by step solution

01

Identify the polynomial and value

The given polynomial is \(f(x) = 2x^2 - 3x - 3\). The value of \(k\) is 2. We need to find \(f(2)\).
02

Set up the synthetic division

Write down the coefficients of the polynomial \(f(x)\), which are 2, -3, and -3. Set up the synthetic division with these coefficients and the value of \(k = 2\) on the left side.
03

Perform synthetic division

1. Bring down the first coefficient (2) to the bottom row.2. Multiply this number by \(k\) (which is 2) and write the result (4) under the second coefficient.3. Add: \(-3 + 4 = 1\).4. Repeat the multiplication: \(1 \times 2 = 2\) and write this under the last coefficient.5. Add: \(-3 + 2 = -1\).
04

Find the remainder

The remainder from the synthetic division step is the last number in the bottom row, which is -1. According to the Remainder Theorem, this remainder is \(f(2)\).

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Key Concepts

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

remainder theorem
The Remainder Theorem is a handy tool in algebra. It allows us to find the remainder of a polynomial when divided by \((x - k)\) quickly, just by evaluating the polynomial at \(x = k\). For a polynomial function \(f(x)\) and a number \(k\), the theorem states that the remainder of the division of \(f(x)\) by \((x - k)\) is simply \(f(k)\).
In our example, the polynomial is \(f(x) = 2x^2 - 3x - 3\). Using the remainder theorem, we found \((f(2))\) through synthetic division. Here, the remainder was \(-1\), so \(f(2) = -1\).
Understanding this theorem simplifies many polynomial calculations, saving time and effort.
polynomial function
A polynomial function is an expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents of variables. Polynomials are written in the form:
\(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\),
where \(a_n, a_{n-1},..., a_1, a_0\) are constants and \(n\) is a non-negative integer.
In our specific exercise, we worked with the polynomial \(f(x) = 2x^2 - 3x - 3\). Here:
  • \(a_2 = 2\)
  • \(a_1 = -3\)
  • \(a_0 = -3\)
These coefficients are essential for performing operations like synthetic division and for understanding the behavior of the polynomial.
evaluating polynomials
Evaluating a polynomial means finding the value of the polynomial function for a specific value of \(x\). This is done by substituting the given value into the polynomial and performing the arithmetic operations.
For example, with \(f(x) = 2x^2 - 3x - 3\) and \((k = 2)\), we wanted to find \(f(2)\). By substituting 2 into the polynomial:\begin{align*}2(2)^2 - 3(2) - 3 = 8 - 6 - 3 = -1\rbrace
Using synthetic division, we also found that \(f(2) = -1\). Both methods confirmed the same result, illustrating the consistency and reliability of polynomial evaluation and the Remainder Theorem.

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Most popular questions from this chapter

Which function 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. \(f(x)=\frac{x+5}{(x+2)(x-3)}\)

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=x^{6}-9 x^{4}-16 x^{2}+144$$

Work each problem.For \(k>0,\) if \(y\) varies directly as \(x,\) then when \(x\) increases, \(y\)_________ and when \(x\) decreases, \(y\)___________.

Solve each problem. AIDS Cases in the United States The table* lists the total (cumulative) number of AIDS cases diagnosed in the United States through \(2007 .\) For example, a total of \(361,509\) AIDS cases were diagnosed through 1993 (a) Plot the data. Let \(x=0\) correspond to the year 1990 . (b) Would a linear or a quadratic function model the data better? Explain. (c) Find a quadratic function defined by \(f(x)=a x^{2}+b x+c\) that models the data. (d) Plot the data together with \(f\) on the same coordinate plane. How well does \(f\) model the number of AIDS cases? (e) Use \(f\) to estimate the total number of AIDS cases diagnosed in the years 2009 and 2010 (f) According to the model, how many new cases were diagnosed in the year \(2010 ?\) $$\begin{array}{c|c||c|c} \text { Year } & \text { AIDS Cases } & \text { Year } & \text { AIDS Cases } \\\ \hline 1990 & 193,245 & 1999 & 718,676 \\ \hline 1991 & 248,023 & 2000 & 759,434 \\ \hline 1992 & 315,329 & 2001 & 801,302 \\ \hline 1993 & 361,509 & 2002 & 844,047 \\ \hline 1994 & 441,406 & 2003 & 888,279 \\ \hline 1995 & 515,586 & 2004 & 932,387 \\ \hline 1996 & 584,394 & 2005 & 978,056 \\ \hline 1997 & 632,249 & 2006 & 982,498 \\ \hline 1998 & 673,572 & 2007 & 1,018,428 \\ \hline \end{array}$$

Solve each problem. AIDS Deaths in the United States The table* lists the total (cumulative) number of known deaths caused by AIDS in the United States up to 2007 (a) Plot the data. Let \(x=0\) correspond to the year 1990 . (b) Would a linear or a quadratic function model the data better? Explain. (c) Find a quadratic function defined by \(g(x)=a x^{2}+b x+c\) that models the data. (d) Plot the data together with \(g\) on the same coordinate plane. How well does \(g\) model the number of AIDS cases? (e) Use \(g\) to estimate the total number of AIDS deaths in the year 2010 . (f) Consider the last two entries in the table for the years 2006 and 2007 . Is it safe to assume that the quadratic model given for \(g(x)\) will continue for years 2008 and beyond? $$\begin{array}{c|c||c|c} \hline \text { Year } & \text { AIDS Deaths } & \text { Year } & \text { AIDS Deaths } \\ \hline 1990 & 119,821 & 1999 & 419,234 \\ \hline 1991 & 154,567 & 2000 & 436,373 \\ \hline 1992 & 191,508 & 2001 & 454,099 \\ \hline 1993 & 220,592 & 2002 & 471,417 \\ \hline 1994 & 269,992 & 2003 & 489,437 \\ \hline 1995 & 320,692 & 2004 & 507,536 \\ \hline 1996 & 359,892 & 2005 & 524,547 \\ \hline 1997 & 381,738 & 2006 & 565,927 \\ \hline 1998 & 400,743 & 2007 & 583,298 \\ \hline \end{array}$$
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