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Chapter 3: Problem 90

Find the conjugate of each number. $$-3+i$$

Short Answer

Expert verified
The conjugate is \(-3-i\).

Step by step solution

01

Understanding Conjugates

The conjugate of a complex number is obtained by changing the sign of the imaginary part. If we have a complex number of the form \(a + bi\), its conjugate will be \(a - bi\). In this case, our number is \(-3 + i\).
02

Identifying Components

Write down the real part and the imaginary part of the given number. For \(-3+i\), the real part is \(-3\) and the imaginary part is \(i\).
03

Applying Conjugate Definition

Change the sign of the imaginary part. So, the imaginary part \(+i\) becomes \(-i\).
04

Writing the Conjugate

Combine the real part and the new imaginary part to form the conjugate. The conjugate of \(-3+i\) is \(-3-i\).

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Key Concepts

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

Conjugate
The concept of a conjugate in complex numbers revolves around altering the sign of the imaginary part of a complex number. When you're given a complex number, say in the form
  • \( a + bi \)
'identifying and changing this imaginary part is how you find its conjugate.
For instance, if you start with
  • \(-3 + i\)
the imaginary portion, \(+i\), is adjusted to \(-i\).
This doesn't affect the real part of the number. Thus, the conjugate becomes
  • \(-3 - i \).
The process involves a simple sign change but plays a critical role in various calculations, such as simplifying expressions or finding reciprocals of complex numbers.
Imaginary Part
The imaginary part of a complex number is what sets it apart from real numbers. In a complex number of the form
  • \( a + bi \)
'b' is the coefficient of the imaginary unit 'i'.
For example, in the complex number
  • \(-3 + i\)
the imaginary part is \(+i\) or \(+1i\).
The imaginary component is crucial when considering operations like finding conjugates or performing arithmetic with complex numbers.
This is because the imaginary part is the piece of the number that gets altered to find the conjugate, making its understanding pivotal.
Real Part
The real part of a complex number is essentially the portion that does not involve the imaginary unit.
  • This part aligns with the numbers you might already know, broadly classed as 'real numbers.'
In a complex number expressed as
  • \( a + bi \)
\('a'\) represents the real part.
Consider the example
  • \(-3 + i\)
, where the real part is clearly
  • \(-3\).
Unlike the imaginary part, the real part remains unchanged when finding the conjugate.
It serves as the constant anchor in complex numbers that provides a reference point around which calculations are performed.

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Most popular questions from this chapter

Use the concepts of this section. Explain why a polynomial function of degree 4 with real coefficients has either zero, two, or four real zeros (counting multiplicities).

Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros for \(P(x) .\) Then, use a graph to determine the actual numbers of positive and negative real zeros. $$P(x)=x^{5}+3 x^{4}-x^{3}+2 x+3$$

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=-3 x^{4}+22 x^{3}-55 x^{2}+52 x-12$$

Solve each problem. Suppose that a person's heart rate, \(x\) minutes after vigorous exercise has stopped, can be modeled by $$f(x)=\frac{4}{5}(x-10)^{2}+80$$ The output is in beats per minute, where the domain of \(f\) is \(0 \leq x \leq 10\) (a) Evaluate \(f(0)\) and \(f(2) .\) Interpret the result. (b) Estimate the times when the person's heart rate was between 100 and 120 beats per minute, inclusive.

Use the concepts of this section. Suppose that \(k, a, b,\) and \(c\) are real numbers, \(a \neq 0,\) and a polynomial function \(P(x)\) may be expressed in factored form as \((x-k)\left(a x^{2}+b x+c\right)\). (a) What is the degree of \(P ?\) (b) What are the possible numbers of distinct real zeros of \(P ?\) (c) What are the possible numbers of nonreal complex zeros of \(P ?\) (d) Use the discriminant to explain how to determine the number and type of zeros of \(P\).
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