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Chapter 16: Problem 17

Determine the location and order of the zeros. $$\left(3 z^{2}+1\right) e^{-2}$$

Short Answer

Expert verified
The zeros are \(i\cdot\sqrt{\frac{1}{3}}\) and \(-i\cdot\sqrt{\frac{1}{3}}\), each with order 1.

Step by step solution

01

Understanding the Given Function

The function given is \((3z^2 + 1)e^{-2}\). The zeros of a function occur where its value is zero.
02

Rewrite & Simplify the Expression

Rewrite the expression \((3z^2 + 1)e^{-2} = 0\). Since \(e^{-2}\) is a non-zero constant, the equation simplifies to \(3z^2 + 1 = 0\).
03

Set Up the Equation to Find Zeros

Set up the equation based on the simplified form: \[3z^2 + 1 = 0\] This equation should solve for \(z\).
04

Solve for Zeros

To find zeros, solve the equation:\[3z^2 = -1\]\[z^2 = -\frac{1}{3}\]
05

Identify Zeros and their Order

Since \(z^2 = -\frac{1}{3}\), the zeros are: \(z = \sqrt{-\frac{1}{3}}\) and \(z = -\sqrt{-\frac{1}{3}}\). These represent complex numbers because the square root of a negative number is imaginary.
06

Determine the Nature of Zeros

Rewrite imaginary roots as \[z = i\cdot\sqrt{\frac{1}{3}}\] and \[z = -i\cdot\sqrt{\frac{1}{3}}\]. The order of zeros is 1 for each root since both come from the same quadratic equation.

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Key Concepts

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

Zeros of a Function
Zeros of a function are points where the function's value is equal to zero. These points are important in understanding the behavior of the function, especially where it crosses the x-axis in real-valued graphs. For a complex function, zeros help us determine where the output is zero, even if it involves imaginary numbers.
To find zeros:
  • Set the function equal to zero.
  • Solve the resulting equation for the variable.
The nature of the zeros, such as whether they are real or complex, depends on the type of equation. Even if a function contains fractions or complex numbers, solving for zeros follows these steps. Understanding the zeros helps to identify the key points of the function, offering insights into its graph and potential intersection points.
Quadratic Equations
Quadratic equations follow the standard form of ax^2 + bx + c = 0, where a, b, and c are constants. These equations may have two real roots, one real root, or two complex roots, depending on the discriminant (b^2 - 4ac).
The discriminant helps to determine the nature of the roots:
  • If it is positive, two distinct real roots exist.
  • If zero, one real root (a repeated root) exists.
  • If negative, two complex roots occur.
For the given equation 3z^2 + 1 = 0, since there is no real solution (as we cannot have a positive value from a non-negative multiplied by a negative), its roots must be complex. This is evident when the discriminant is negative, leading to imaginary solutions. Quadratic equations provide foundational concepts in solving complex number equations.
Imaginary Numbers
Imaginary numbers offer solutions when equations involve taking the square root of negative numbers. The basic imaginary unit is represented by \(i\), where \(i^2 = -1\). These numbers, used alongside real numbers, form complex numbers, which are written as a + bi where a is the real part and bi is the imaginary part.
Understanding imaginary numbers involves:
  • Recognizing when roots of a negative number are needed.
  • Using the substitution \(i = \sqrt{-1}\).
In the problem given, the zeros of the quadratic equation are expressed in terms of imaginary numbers. They are \(z = \pm i \cdot \sqrt{\frac{1}{3}}\), showcasing how imaginary numbers offer meaningful solutions in mathematical contexts where real numbers do not suffice.

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