What Are the Values of r and s in the Polynomial q(z) with Given Roots?

In summary, the polynomial q(z) = z^3 − z^2 + rz + s has complex roots 1 + i and i. The third complex root is -2i. To find r and s, we can use the same method as in the previous problem. The sum of the roots must be real, so the third root must contain -2i.
  • #1
53Mark53
52
0

Homework Statement


[/B]
Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero.

The Attempt at a Solution


[/B]
(z-(1+i)(z-i) = Z^2-z-1-2iz+i

(Z^2-z-1-2iz+i)(z+d) = Z^3+z^2(d-1-zi)-z(d+1+2di-i)-d(1-i)

Z^2 term

Z^2(d-1-zi)=-z^2
d-1-2i=-1
d=2i

z term

-z(d+1+2di-i)=rz
-d-1-2di+i=r
2i-1-4i^2+i=r
-i+3=r

constant term

-d(1-i)=s
-2i+2i^2=s
-2i-2=s

This is what I have done but I am when I expand the complex zeros I do not get anything close to q(z)
 
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  • #2
For start I believe you do something wrong with the factor ##(z+d)## in line 2 of your post, shouldn't that be ##(z-d)##?
 
  • #3
Delta² said:
For start I believe you do something wrong with the factor ##(z+d)## in line 2 of your post, shouldn't that be ##(z-d)##?
Why would it matter if it is positive or negative?
 
  • #4
It matters on the sign d will have. I believe the 3rd root of your polynomial is not 2i but -2i.

if you write a polynomial as ##p(z)=(z+d)q(z)## then one of the roots of ##p(z)## is not d but -d as you can easily verify.
 
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  • #5
Delta² said:
It matters on the sign d will have. I believe the 3rd root of your polynomial is not 2i but -2i.

if you write a polynomial as ##p(z)=(z+d)q(z)## then one of the roots of ##p(z)## is not d but -d as you can easily verify.
Thanks that works :)
 
  • #6
53Mark53 said:

Homework Statement


[/B]
Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero.

I think you can achieve some shortcuts here. Equation is of exactly the same form as one for a problem you have just solved (or claimed to :oldwink: ). Help with finding Zeros of a polynomial with 1+i as a zero

In the previous equation you were given two roots and found the third, or somebody did.

Here you are given two roots which are the same as two of the roots of the previous problem multiplied by -i if I am not mistaken.

If all roots of a polynomial are multiplied by the same factor, what happens to the coefficients?

Edit: however that does not seem to work in the way I guessed. :redface:

The sum of roots must be real, so the third root must contain -2i.

The problem can be done in the same way as before.

There ought to be some greater analogy with the previous problem, which at the moment I cannot see.

 
Last edited:

1. What is a complex polynomial?

A complex polynomial is a mathematical expression consisting of a sum of terms, each of which is a product of a constant and one or more variables raised to non-negative integer powers. The coefficients and variables can be complex numbers, which are numbers that contain both a real and imaginary component.

2. How do you find the degree of a complex polynomial?

The degree of a complex polynomial is determined by the highest power of the variable in the expression. For example, in the polynomial 3x2 + 4x + 2, the degree is 2. If the polynomial is in factored form, the degree is the sum of the exponents of the variables in the highest order term.

3. Can a complex polynomial have more than one variable?

Yes, a complex polynomial can have more than one variable. In fact, it is common for complex polynomials to have multiple variables, such as in the expression 2x2y + 5xy2 + 3y3. In this case, the degree of the polynomial is determined by the highest total degree of the variables, which in this case is 3.

4. How do you solve a complex polynomial?

To solve a complex polynomial, you can use various methods such as factoring, the quadratic formula, or synthetic division. The specific method used will depend on the degree and complexity of the polynomial. It is also important to remember to find all possible solutions, including complex solutions, when solving a complex polynomial.

5. What is the fundamental theorem of algebra and how does it relate to complex polynomials?

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients has at least one complex root. This means that every complex polynomial can be factored into linear and quadratic factors, which can then be solved to find all possible complex solutions. This theorem is essential in solving and understanding complex polynomials.

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