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Chapter 11: Problem 40

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

Short Answer

Expert verified
The value of \ f(3 - 2i) \ is \ 5 - 10i \.

Step by step solution

01

Substitution and the Remainder Theorem

According to the remainder theorem, if a polynomial function is divided by \(x - k\), the remainder is \(f(k)\). To find \(f(3 - 2i)\) in \(f(x) = x^2 - x + 3\), substitute \(3 - 2i\) into the polynomial.
02

Substitute \(3 - 2i\) into the polynomial

Substitute \(k = 3 - 2i\) into \(f(x)\): \ f(3 - 2i) = (3 - 2i)^2 - (3 - 2i) + 3 \.
03

Expand and Simplify

First, calculate \( (3 - 2i)^2 \): \ (3 - 2i)^2 = 9 - 12i + 4i^2 \ (since \(i^2 = -1\)) \ = 9 - 12i - 4 = 5 - 12i \. Then substitute and simplify: \ (3 - 2i)^2 - (3 - 2i) + 3 = (5 - 12i) - (3 - 2i) + 3 \. Combine like terms: \ (5 - 12i) - 3 + 2i + 3 = 5 - 3 + 3 - 12i + 2i = 5 - 3 + 3 - 10i \. Thus, \ f(3 - 2i) = 5 - 12i - 3 + 2i + 3 \.
04

Final Simplification

Combine the real and imaginary parts: \(5 - 3 + 3 - 10i = 5 \- 10i\). Thus, \( f(3 - 2i) = 5 - 10i \).

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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, structured in a specific format. A general polynomial function can be written as:
\(f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \), where: \begin{itemize} \item \text{ n is the degree of the polynomiala_{i} \ a_n a_0 are the coefficients \ (constants)\} is \} \ \

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Most popular questions from this chapter

Determine whether each polynomial function is even, odd, or neither. \(f(x)=0.2 x^{4}\)

Use synthetic division to divide. $$ \frac{3 x^{5}+x^{4}-84 x^{2}-12 x+3}{x-3} $$

Approximate to the nearest hundredth the coordinates of the turning point in the given interval of the graph of each polynomial function. \(f(x)=2 x^{3}-5 x^{2}-x+1, \quad[1.4,2]\)

Use a graphing calculator to find (or approximate) the real zeros of each function \(f(x)\). Express decimal approximations to the nearest hundredth. \(f(x)=4 x^{4}+8 x^{3}-4 x^{2}+4 x+1\)

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
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