Monic polynomial of the lowest possible degree

In summary, for a real monic polynomial with zeros −1−2i,−2i and i, the polynomial of the lowest possible degree is z^3 + z^2 + 4z + 4. This is found by expanding the zeros (z+1+2i)(z+2i)(z-i) and making sure to include the conjugates of the original roots. The conjugates of complex numbers a + bi is a - bi. Thus, the conjugates of the original roots are -1+2i, -2i, and -i. The polynomial is then simplified to z^3 + z^2 + 4z + 4, and the real zeros are -1
  • #1
53Mark53
52
0

Homework Statement



A monic polynomial is a polynomial which has leading coefficient 1. Find the real, monic polynomial of the lowest possible degree which has zeros −1−2i,−2i and i. Use z as your variable.

The Attempt at a Solution


[/B]
Would I just expand the zeros giving me

(z+1+2i)(z+2i)(z-i)

z^3+(1+3*I)*z^2+I*x+I*4+2
 
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  • #2
53Mark53 said:

Homework Statement



A monic polynomial is a polynomial which has leading coefficient 1. Find the real, monic polynomial of the lowest possible degree which has zeros −1−2i,−2i and i. Use z as your variable.

The Attempt at a Solution


[/B]
Would I just expand the zeros giving me

(z+1+2i)(z+2i)(z-i)

z^3+(1+3*I)*z^2+I*x+I*4+2
Don't forget: It's a real polynomial.
 
  • #3
SammyS said:
Don't forget: It's a real polynomial.
how would I make it real?
 
  • #4
53Mark53 said:
how would I make it real?
Hint: what happened to the conjugates of the original roots?
 
  • #5
SteamKing said:
Hint: what happened to the conjugates of the original roots?
SteamKing said:
Hint: what happened to the conjugates of the original roots?

Does that mean that I have to square the imaginary roots?
 
  • #6
53Mark53 said:
Does that mean that I have to square the imaginary roots?
What is the conjugate of the complex number, a + bi?
 
  • #7
@53Mark53 ,
Here's what you said in a previous thread:
53Mark53 said:

Homework Statement


p(x) = x^3 − x^2 + ax + b is a real polynomial with 1 + i as a zero, find a and b and find all of the real zeros of p(x).

The Attempt at a Solution


[/B]
1-i is also a zero as it is the conjugate of 1+i
The same applies here, but now you have 3 complex roots, no two of which form a conjugate pair.
 
  • #8
SammyS said:
@53Mark53 ,
Here's what you said in a previous thread:

The same applies here, but now you have 3 complex roots, no two of which form a conjugate pair.
Thanks I got the right answer by using conjugate pairs
 

Related to Monic polynomial of the lowest possible degree

1. What is a monic polynomial of the lowest possible degree?

A monic polynomial of the lowest possible degree is a polynomial in which the coefficient of the highest degree term is equal to 1.

2. How do you determine the lowest possible degree of a monic polynomial?

The lowest possible degree of a monic polynomial can be determined by looking at the highest degree term and making sure its coefficient is equal to 1. If it is not, the polynomial can be simplified to achieve the lowest possible degree.

3. Can a monic polynomial have a degree of 0?

Yes, a monic polynomial can have a degree of 0 if it is a constant term with the value of 1. In this case, the polynomial would be written as f(x) = 1.

4. How is a monic polynomial different from a regular polynomial?

A monic polynomial is different from a regular polynomial because it has a coefficient of 1 on the highest degree term. Regular polynomials can have any coefficient on the highest degree term.

5. Why is it important to have a monic polynomial of the lowest possible degree?

Having a monic polynomial of the lowest possible degree can help simplify and solve polynomial equations more easily. It can also make it easier to identify the degree and leading coefficient of the polynomial.

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