Chapter 14: Problem 33
Find out how many roots the equations in problem have in each quadrant. \(z^{6}+z^{3}+9 z+64=0\) (no real roots)
Short Answer
Expert verified
Given symmetrical roots: First Quadrant 2, Second Quadrant 2, Third Quadrant 1, Fourth Quadrant 1.
Step by step solution
01
Understand the Problem
The equation to solve is a polynomial equation of degree 6, given by \[z^{6}+z^{3}+9z+64=0\]with the constraint that there are no real roots.
02
Analyze Complex Roots
Since the polynomial has no real roots, all roots must be complex. Recall that complex roots come in conjugate pairs for polynomials with real coefficients, meaning if \(a + bi\) is a root, then \(a - bi\) is also a root.
03
Count the Number of Roots
Given it's a 6th degree polynomial, there are exactly 6 roots.
04
Locate the Quadrants
Complex roots in the form \(a + bi\) and \(a - bi\) are symmetrically located around the real axis. Since each complex root pair corresponds to different quadrants, count how pairs are distributed.
05
Distribute the Roots
If a root pair is \(a + bi\) and \(a - bi\):
06
Step 5.1: First Quadrant
Roots of the form \(a + bi\) with \(b > 0, a > 0\) (First Quadrant), its conjugate root \(a - bi\) will be in the Fourth Quadrant.
07
Step 5.2: Second Quadrant
Roots of the form \(-a + bi\) with \(b > 0, a > 0\) (Second Quadrant), its conjugate root \(-a - bi\) will be in the Third Quadrant.
08
Step 5.3: Third and Fourth Quadrants
For every pair in the first and second quadrants, their conjugate pairs will be in the fourth and third quadrants respectively.
09
Conclusion
Polynomials of degree 6 with no real roots and symmetric complex conjugate pairs mean we have an equal or symmetrical distribution. So, in each pair, respectively split among quadrants equally.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Roots
A polynomial is an algebraic expression that involves variables and coefficients, connected by addition, subtraction, multiplication, and non-negative integer exponents. When solving polynomial equations such as \(z^6 + z^3 + 9z + 64 = 0\), we're looking for values of the variable (in this case, \(z\)) that make the equation true. These values are called the 'roots' of the polynomial. For a polynomial equation of degree \(n\), there are exactly \(n\) roots. Understanding this principle helps in figuring out how many roots you will be dealing with.
Complex Conjugate Pairs
When dealing with polynomials with real coefficients, complex roots always come in conjugate pairs. This means if \(a + bi\) is a root, \(a - bi\) will also be a root. Here's why: If you have a polynomial with real coefficients, all the imaginary parts (\(i\)) must cancel out to result in an equation that holds true (since real coefficients need to produce real results). Conjugate pairs ensure this balance, as the imaginary parts cancel each other. Therefore, for our polynomial \(z^6 + z^3 + 9z + 64 = 0\), the six roots will be grouped in three pairs of conjugates. Each pair will consist of roots like \(a + bi\) and \(a - bi\).
Quadrants
In the complex plane, each complex number corresponds to a point, represented as \(a + bi\). The plane is divided into four quadrants:
- **First Quadrant:** \(a > 0, b > 0\)
- **Second Quadrant:** \(a < 0, b > 0\)
- **Third Quadrant:** \(a < 0, b < 0\)
- **Fourth Quadrant:** \(a > 0, b < 0\)
Since complex conjugate pairs are symmetric with respect to the real axis, if one root is in the First Quadrant, its conjugate will be in the Fourth Quadrant. Similarly, if one is in the Second Quadrant, its conjugate will be in the Third Quadrant. For our polynomial with three pairs of complex conjugates, each pair will distribute roots across these four quadrants equally. So, we would have each pair taking up two quadrants symmetrically.
- **First Quadrant:** \(a > 0, b > 0\)
- **Second Quadrant:** \(a < 0, b > 0\)
- **Third Quadrant:** \(a < 0, b < 0\)
- **Fourth Quadrant:** \(a > 0, b < 0\)
Since complex conjugate pairs are symmetric with respect to the real axis, if one root is in the First Quadrant, its conjugate will be in the Fourth Quadrant. Similarly, if one is in the Second Quadrant, its conjugate will be in the Third Quadrant. For our polynomial with three pairs of complex conjugates, each pair will distribute roots across these four quadrants equally. So, we would have each pair taking up two quadrants symmetrically.