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Chapter 2: Problem 11

A polynomial function is described. Find all remaining zeros. Degree: \(4 \quad\) Zeros: \(2 i, 3-i\)

Short Answer

Expert verified
All zeros are: \(2i, -2i, 3-i, 3+i\).

Step by step solution

01

Understand the Conjugate Pairs Theorem

Complex zeros in polynomials with real coefficients occur in conjugate pairs. Since the given zeros include \(2i\) and \(3 - i\), we can conclude that their conjugates, \(-2i\) and \(3 + i\), must also be zeros of the polynomial.
02

Count the Zeros

We know the polynomial is of degree 4, meaning it can have exactly 4 zeros (considering multiplicity). Since we already have four zeros: \(2i, -2i, 3-i,\) and \(3+i\), we have accounted for all the zeros.

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Key Concepts

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

Conjugate Pairs
In the context of polynomial functions with real coefficients, conjugate pairs play a crucial role. When a polynomial has a complex zero, such as the given zero
  • \(2i\),
  • \(3 - i\),
its conjugate is also a zero. The conjugate of a complex number
  • \(a + bi\)
  • \(a - bi\)
where
  • "a" and "b" are real numbers, and "\(i\)" is the imaginary unit with the property \(i^2 = -1\).
This means if
  • \(2i\) is a zero,
  • \(-2i\) must also be a zero.
  • Similarly,
  • if \(3 - i\) is a zero,
  • then \(3 + i\) is also a zero.
This theorem of conjugate pairs ensures that the polynomial maintains real coefficients.
Degree of Polynomial
The degree of a polynomial is a vital concept in understanding its nature. It indicates the highest power of the variable present in the polynomial. For example, if a polynomial is of degree 4, the term with the highest power of the variable will be raised to the fourth power.
  • In this exercise, the polynomial's degree is given as 4.
This informs us that the polynomial can have up to 4 zeros, assuming we count the multiplicity of each zero.
  • Such a property is significant because it limits the total number of roots (or zeros) a polynomial can have.
  • If it has complex roots, their number will match the degree as they pair off with their conjugates wisely.
Understanding the degree helps in constructing and understanding the behavior of polynomial functions.
Complex Numbers
Complex numbers are a core concept when analyzing polynomials, especially when dealing with zeros that are not real numbers. A complex number has a real part and an imaginary part, expressed generally as \(a + bi\), where
  • "a" is the real component,
  • "b" is the coefficient for the imaginary component,
  • and \(i\) is the imaginary unit such that \(i^2 = -1\).
In this context, you might encounter complex zeros such as
  • \(2i\)
  • \(3 - i\)
These zeros are significant because they influence the polynomial's roots and make it necessary to consider their conjugates. Understanding complex numbers can help solve polynomial equations, especially when the polynomials do not have all real solutions.
Multiplicity of Roots
Multiplicity refers to the number of times a particular root is a zero for a given polynomial. For example, if a zero appears more than once, it has multiplicity greater than one. This concept is essential when dealing with the degree of a polynomial, as the sum of all multiplicities equals the polynomial's degree.
  • In the given exercise, since the polynomial's degree is 4, the total multiplicities combined should equal 4.
The roots
  • \(2i\),
  • \(-2i\),
  • \(3 - i\),
  • and \(3 + i\)
all have multiplicity 1 in this case because they each appear only once. Understanding multiplicity helps in solving polynomial equations correctly, ensuring all zeros are accounted for.

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