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Chapter 2: Problem 46

Finding a Polynomial Function with Given Zeros, find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$4,-3 i$$

Short Answer

Expert verified
The polynomial function with real coefficients that has the given zeros is \(P(x) = x^3 - 4x^2 + 9x - 36\).

Step by step solution

01

Determine the Conjugate Pair

Given zeros are 4 and \(-3i\). Since our polynomial has real coefficients and we have a complex root, the conjugate \(3i\) must also be a root.
02

Write in Factored Form

The polynomial P(x) can be written in factored form as P(x) = \(a(x - r_1)(x - r_2)(x - r_3)\) etc, where a is a nonzero constant, and \(r_1, r_2, r_3\) etc are the roots. Therefore, \(P(x) = a(x - 4)(x - (-3i))(x - 3i)\).
03

Simplify the Polynomial

Simplify \(P(x) = a(x - 4)(x - (-3i))(x - 3i) = a(x - 4)(x^2 + 9)\) assuming \(a = 1\) for the simplest form of the polynomial. Therefore, \(P(x) = x^3 - 4x^2 + 9x - 36\).

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Key Concepts

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

Zeros of Polynomials
Understanding the zeros of a polynomial is crucial in algebra. Zeros are the values of variable \( x \) that make the polynomial equal to zero. In simpler terms, they're where the graph of the polynomial hits the x-axis.
  • If you have a polynomial such as \( P(x) = x^3 - 4x^2 + 9x - 36 \), you can find its zeros by setting \( P(x) = 0 \) and solving for \( x \).
  • Zeros can be real numbers (like 4 in our original exercise) or complex numbers (like \(-3i\)).
  • When a polynomial has real coefficients and includes complex zeros, those zeros always appear in conjugate pairs.
Thus, understanding zeros informs the behavior of the polynomial and helps in constructing the polynomial from known zeros.
Complex Conjugate Roots
Complex conjugate roots are essential when dealing with polynomials with real coefficients. If a polynomial equation with real coefficients has a complex root, its complex conjugate is also a root.
  • This means if \(-3i\) is a root, then \(3i\) must also be a root.
  • Complex conjugates like \(-3i\) and \(3i\) come in pairs to ensure the coefficients remain real.
For example, for the polynomial in our exercise, given one complex root at \(-3i\), we automatically include \(3i\) as another root. This concept ensures every polynomial with real coefficients remains invariant of imaginary numbers alone and contains balanced, real, and imaginary sets of roots.
Factored Form of Polynomials
Placing a polynomial in factored form helps visualize and compute its structure effectively. Here's how it works:
  • The basic form is \( P(x) = a(x - r_1)(x - r_2)...(x - r_n) \), where \( a \) is a constant and \( r_1, r_2, ..., r_n \) are the zeros.
  • In our example, the roots are \( 4, -3i, \) and \( 3i \), so the factored form starts as \( P(x) = a(x - 4)(x - (-3i))(x - 3i) \).
  • Simplifying this further while setting \( a = 1 \) for ease gives us \( P(x) = (x - 4)(x^2 + 9) \).
Factored form provides a straightforward path for expansion or further manipulation and allows easy identification of the zeros. This form is especially crucial for polynomial manipulation and solving high-order polynomial equations.

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Most popular questions from this chapter

Forensics At \(8 : 30\) A.M., a coroner went to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At \(9 : 00\) A.M. the temperature was \(85.7^{\circ} \mathrm{F},\) and at \(11 : 00\) A.M. thetemperature was \(82.8^{\circ} \mathrm{F}\) . From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula $$t=-10 \ln \frac{T-70}{98.6-70}$$ where \(t\) is the time in hours elapsed since the person died and \(T\) is the temperature (in degrees Fahrenheit) of the person's body. (This formula comes from a general cooling principle called Newton's Law of Cooling.It uses the assumptions that the person had a normal body temperature of \(98.6^{\circ} \mathrm{F}\) at death and that the room temperature was a constant \(70^{\circ} \mathrm{F} .\) ) Use the formula to estimate the time of death of the person.

Finding the Domain of an Expression In Exercises \(61-66\) , find the domain of the expression. Use a graphing utility to verify your result. $$\sqrt{81-4 x^{2}}$$

Depreciation A laptop computer that costs \(\$ 1150\) new has a book value of \(\$ 550\) after 2 years. (a) Find the linear model \(V=m t+b .\) (b) Find the exponential model \(V=a e^{k t} .\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.

Graphical Analysis In Exercises \(63-66,\) use a graphing utility to graph the rational function. State the domain of the function and find any asymptotes. Then zoom out sufficiently far so that the graph appears as a line. Identify the line. $$f(x)=\frac{x^{2}+5 x+8}{x+3}$$

A Rational Function with a Slant Asymptote In Exercises \(49-62,\) (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical or slant asymptotes, and (d) plot additional solution points as needed to sketch the graph of the rational function. $$f(x)=\frac{2 x^{3}+x^{2}-8 x-4}{x^{2}-3 x+2}$$
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