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Chapter 11: Problem 43

Find a polynomial function \(f(x)\) of least possible degree with only real coefficients and having the given zeros. \(1+2 i, 2\) (multiplicity 2)

Short Answer

Expert verified
The polynomial function is \(f(x) = x^4 - 6x^3 + 17x^2 - 28x + 20\).

Step by step solution

01

Identify all zeros

The given zeros of the polynomial are: 1. Complex zero: \(1 + 2i\) 2. Real zero with multiplicity 2: \(2\)
02

Include the conjugate of the complex zero

Since polynomial functions with real coefficients must have complex zeros occur in conjugate pairs, the complex conjugate of \(1 + 2i\) is \(1 - 2i\). So, the zeros are \(1 + 2i\), \(1 - 2i\), and \(2\) (with multiplicity 2).
03

Write the polynomial as a product of factors

Express the polynomial in factored form using its zeros: \[ f(x) = (x - (1 + 2i))(x - (1 - 2i))(x - 2)^2 \]
04

Expand the factors involving the complex zeros

First, expand the factors involving the complex zeros: \[ (x - (1 + 2i))(x - (1 - 2i)) = ((x - 1) - 2i)((x - 1) + 2i) \]This is a difference of squares: \[ = (x - 1)^2 - (2i)^2 \]Simplify using \(i^2 = -1\): \[ = (x - 1)^2 - 4(-1) = (x - 1)^2 + 4 = x^2 - 2x + 1 + 4 = x^2 - 2x + 5 \]
05

Multiply the expanded factors

Now, multiply the factor \(x^2 - 2x + 5\) by \((x - 2)^2\): \[ f(x) = (x^2 - 2x + 5)(x - 2)^2 \]First, expand \((x - 2)^2\): \[ (x - 2)^2 = x^2 - 4x + 4 \]Now, distribute \(x^2 - 2x + 5\) over \(x^2 - 4x + 4\): \[ f(x) = (x^2 - 2x + 5)(x^2 - 4x + 4) \]Multiply each term:\[ f(x) = x^2 (x^2 - 4x + 4) - 2x (x^2 - 4x + 4) + 5 (x^2 - 4x + 4) \]\[ = x^4 - 4x^3 + 4x^2 - 2x^3 + 8x^2 - 8x + 5x^2 - 20x + 20 \]Combine like terms:\[ = x^4 - 6x^3 + 17x^2 - 28x + 20 \]

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Key Concepts

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

complex zeros
Complex zeros are the roots of polynomial equations that are not real numbers. They come in pairs called conjugates if the polynomial has real coefficients. If a polynomial has a complex zero like \(1+2i\), it must also have \(1-2i\) to maintain real coefficients. This ensures that when we multiply the factors involving these zeros, the imaginary parts cancel out.
factored form
The factored form of a polynomial is an expression where the polynomial is written as a product of its zero factors. For the given exercise, the zeros are \(1+2i\), \(1-2i\), and \(2\) (with multiplicity 2), so the factored form is:
\[ f(x) = (x - (1 + 2i))(x - (1 - 2i))(x - 2)^2 \].
This breaks the polynomial into simpler chunks, making it easier to study its properties and graph.
multiplicity of zeros
The multiplicity of a zero refers to how many times a particular zero appears in the polynomial. In the given problem, the zero \(2\) has a multiplicity of 2, meaning it's a double root. This affects the shape of the polynomial graph: near zero with greater multiplicity, the curve flattens out.
Both higher and lower multiplicities of zeros play a significant role in the polynomial's overall behavior and roots.
expanding polynomials
Expanding polynomials means multiplying the factors together to get a single polynomial expression.
Starting with:
\((x - (1+2i))(x - (1-2i))(x - 2)^2\)
Expand the complex factors using the difference of squares:
\((x-1)^2 - (2i)^2\) which simplifies to \(x^2 - 2x + 5\).
Next, multiply by \( (x - 2)^2 \), which is expanded as \(x^2 - 4x + 4\).
Finally, multiply these to get expanded polynomial:
\ f(x) = (x^2 - 2x + 5)(x^2 - 4x + 4)\
\ = x^4 - 6x^3 + 17x^2 - 28x + 20\ Expanding is useful for solving equations, derivatives, and integrations related to polynomials.

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Most popular questions from this chapter

A rectangular piece of cardboard measuring 12 in. by 18 in. is to be made into a box with an open top by cutting equal-size squares from each corner and folding up the sides. Let \(x\) represent the length of a side of each such square in inches. Give approximations to the nearest hundredth. (a) Give the restrictions on \(x\). (b) Determine a function \(V\) that gives the volume of the box as a function of \(x\). (c) For what value of \(x\) will the volume be a maximum? What is this maximum volume? (Hint: Use the function of a graphing calculator that enables us to determine a maximum point within a given interval.) (d) For what values of \(x\) will the volume be greater than 80 in. \(^{3}\) ?

Use synthetic division to divide. $$ \frac{4 x^{2}+19 x-5}{x+5} $$

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.

For each polynomial function, use the remainder theorem and synthetic division to find \(f(k) .\) $$ f(x)=x^{2}-4 x+5 ; \quad k=3 $$

Express each polynomial function in the form \(f(x)=(x-k) q(x)+r\) for the given value of k. $$ f(x)=-2 x^{3}+6 x^{2}+5 ; \quad k=2 $$
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