(2+i) is a root of x^4-5x^3+2x^2-9x+1 determine other root

Question

how do you determine the other root?

Answer 1

This is a quartic function (the highest power is to the fourth ^4)

http://en.wikipedia.org/wiki/Degree_of_a_polynomial#Names_of_polynomials_by_degree

There are as many roots as the power, in this case four. 

The roots are

x=.11317

x=4.9547

 

x=-.03395-1.33501*i

x=-.03395+1.33501*i

-the complex roots follow the rule that the other comment noted, where if one is a+bi the other must be a-bi

Answer 2

Complex roots always come in conjugate pairs so 2 - i is also a root. 

You can then write out the quadratic equation these 2 roots came from using the fact that if you know the 2 roots, the general form of the quadratic equation they came from is 

 

x^2 - (sum of roots ) x  + (product of roots) = 0    Your roots add to 4 and multiply to 5 so your equation is x^2 - 4x +5 = 0. 

 

Now there's a problem. It turns out that 2+i isnt even a root. Actually, a root is the answer to an equation and you dont even have an =. I assumed that you meant = 0, but then 2+i isnt a root.

 

In any case, if 2+i were a root, 2-i MUST also be a root.