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Chapter 3: Problem 6

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{5}+9 x^{3}$$

Short Answer

Expert verified
Zeros: 0 (triple), 3i, -3i; Factored form: \(x^3(x - 3i)(x + 3i)\).

Step by step solution

01

Identify Common Factor

First, observe that the terms of the polynomial \(x^5 + 9x^3\) share a common factor. Factor out \(x^3\) from the polynomial: \[P(x) = x^3(x^2 + 9)\]
02

Solve for Real Zeros

To find the zeros of \(P(x)\), set each factor equal to zero. Start with solving \(x^3 = 0\). This gives us the real zeros:\[x^3 = 0 \Rightarrow x = 0\]
03

Solve for Complex Zeros

Next, solve for the zeros of the quadratic factor: \(x^2 + 9 = 0\). Subtract 9 from both sides:\[x^2 = -9\]Take the square root of both sides to find the complex zeros:\[x = \pm 3i\]
04

List All Zeros

Having derived the solutions from Steps 2 and 3, list all zeros of the polynomial:\(x = 0\) (repeated three times), \(x = 3i\), and \(x = -3i\).
05

Factor the Polynomial Completely

Use the zeros to write the polynomial as a product of its factors:\[P(x) = x^3(x - 3i)(x + 3i)\]This expresses the polynomial completely factored over the complex numbers.

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Key Concepts

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

Real and Complex Zeros
When dealing with polynomials, finding zeros is a crucial step. Zeros of a polynomial, also called roots, are the values for which the polynomial evaluates to zero. In this exercise, we dealt with the polynomial \( P(x) = x^5 + 9x^3 \). Zeros can be either real or complex, and understanding these is fundamental in algebra.
  • Real Zeros: Real zeros are the solutions to the polynomial equation that are real numbers. They graphically represent the points where the polynomial curve intersects the x-axis.
  • Complex Zeros: These are zeros that include imaginary numbers, involving the imaginary unit \( i \) where \( i^2 = -1 \). They do not appear on the real plane in a graph but are vital in expressing polynomials in their simplest form.
In our problem, the real zero is found by solving \( x^3 = 0 \), resulting in \( x = 0 \), whereas the complex zeros are found by solving \( x^2 + 9 = 0 \), giving us \( x = \pm 3i \).
Polynomial Roots
Understanding polynomial roots is the key to solving polynomial equations. In essence, roots are the x-values that make the whole polynomial equal zero. For any given polynomial of degree \( n \), there are at most \( n \) roots. This includes both real and complex zeros.
  • Multiplicity of Roots: A root that occurs more than once in the solution set has a multiplicity greater than one. In our problem, \( x = 0 \) is a root of multiplicity three, meaning it appears three times.
  • Graphical Representation: Roots influence how the graph behaves. For example, a root with odd multiplicity will cross the x-axis, whereas a root with even multiplicity will touch the x-axis and bounce back.
In the given polynomial \( P(x) = x^5 + 9x^3 \), the root \( x = 0 \) touches and crosses the x-axis multiple times, while the complex roots \( x = 3i \) and \( x = -3i \) do not affect the real plot but are critical for factorization.
Factorization Techniques
Polynomials can be factored into simpler components that are easier to work with. Factorization converts a polynomial into a product of its factors, which could either be numbers, variables, or both.
  • Common Factor Extraction: Always check for common factors in the terms of a polynomial. In \( P(x) = x^5 + 9x^3 \), \( x^3 \) is the common factor extracted first, simplifying the expression to \( x^3(x^2 + 9) \).
  • Solving Quadratics: Once simplified, solving remaining simple quadratics or binomials can give further zeros. For example, \( x^2 + 9 \) was solved next to find complex zeros.
  • Complete Factorization: By using all derived factors and zeros, the polynomial can be completely factored. This enables us to write \( P(x) \) as \( x^3(x - 3i)(x + 3i) \), fully expressing the polynomial.
Factorization is essential not only for finding zeros but also for simplifying polynomial expressions and solving polynomial equations efficiently.

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Most popular questions from this chapter

So far, we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. Find all solutions of the equation. (a) \(2 x+4 i=1\) (b) \(x^{2}-i x=0\) (c) \(x^{2}+2 i x-1=0\) (d) \(i x^{2}-2 x+i=0\)

Evaluate the radical expression and express the result in the form \(a+b i\) $$(3-\sqrt{-5})(1+\sqrt{-1})$$

In this chapter we adopted the convention that in rational functions, the numerator and denominator don't share a common factor. In this exercise we consider the graph of a rational function that does not satisfy this rule. (a) Show that the graph of $$ r(x)=\frac{3 x^{2}-3 x-6}{x-2} $$ is the line \(y=3 x+3\) with the point \((2,9)\) removed. [Hint: Factor. What is the domain of \(r ?]\) (b) Graph the rational functions: $$ \begin{aligned} s(x) &=\frac{x^{2}+x-20}{x+5} \\ t(x) &=\frac{2 x^{2}-x-1}{x-1} \\ u(x) &=\frac{x-2}{x^{2}-2 x} \end{aligned} $$

Suppose that the equation \(a x^{2}+b x+c=0\) has real coefficients and complex roots. Why must the roots be complex conjugates of each other? (Think about how you would find the roots using the Quadratic Formula.)

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{3}+4}{2 x^{2}+x-1}$$
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