duki said:Homework Statement
Find a polynomial with integer coefficient for which [tex]\sqrt(2) + i[/tex] is a zero.
Homework Equations
The Attempt at a Solution
I'm not sure where to really start with this one. It is on my review sheet, and I can't remember how to do it. Could someone give me a hand?
To find a polynomial with integer coefficients and a root of √2 + i, first consider that the conjugate of √2 + i is √2 - i. This means that a polynomial with a root of √2 + i will also have a root of √2 - i. From there, you can use the quadratic formula to find the polynomial, which will have the form x2 - 2√2x + 3.
Yes, polynomials with integer coefficients can have complex roots. In fact, any polynomial with degree n will have n complex roots, which may or may not be distinct. This includes polynomials with real and imaginary coefficients.
A polynomial has integer coefficients if all of the terms in the polynomial have only whole number coefficients. To verify this, you can check each term in the polynomial and make sure that all of its coefficients are integers. If even one term has a non-integer coefficient, the polynomial does not have integer coefficients.
Finding polynomials with integer coefficients and a root of √2 + i has several implications. It can help us understand the properties of complex numbers, and it is also useful in solving certain types of equations and in applications such as signal processing and engineering.
Yes, polynomials can have multiple roots of √2 + i. In general, a polynomial of degree n can have n distinct complex roots, which may or may not be repeated. In the case of √2 + i, its conjugate √2 - i is also a root, making it a repeated root.