Don't forget: It's a real polynomial.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
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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
how would I make it real?SammyS said:Don't forget: It's a real polynomial.
Hint: what happened to the conjugates of the original roots?53Mark53 said:how would I make it real?
SteamKing said:Hint: what happened to the conjugates of the original roots?
SteamKing said:Hint: what happened to the conjugates of the original roots?
What is the conjugate of the complex number, a + bi?53Mark53 said:Does that mean that I have to square the imaginary roots?
The same applies here, but now you have 3 complex roots, no two of which form a conjugate pair.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
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1-i is also a zero as it is the conjugate of 1+i
Thanks I got the right answer by using conjugate pairsSammyS 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.
A monic polynomial of the lowest possible degree is a polynomial in which the coefficient of the highest degree term is equal to 1.
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.
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.
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.
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.