Expert: Sherman D. - 3/7/2005

Question
HOw many possible real roots are there for x^5 - 4x^4 + 3x^3 - 2x + 2.
My teacher said that there are either 5,4,3,2,1, or 0 possible roots then he said something like that there can't be 4, 2 or 0 real roots because of something like if there are 4 real roots, it needs 1 imaginary and we know that you can't have 1 imaginary because they come in conjugates.
I don't understand why if you would have 4 real roots, you would need 1 imaginary.Can you please explain?Thank you so much.

Answer
If you go to www.quickmath.com, you will find out that this problem factors to

(x - 1)(x^4 - 3x^3 - 2)

Also if you go to www.quickmath.com and click on solve under equations, you will get 3 real roots and 2 complex ones, but the other 2 real roots are too complex to give in factored form.

Majority of the time, the highest exponent is how many roots you will have.

so you can only have 3 real roots and 2 imaginary ones.

We know it can't be higher than 5, since the highest exponent is 5, and you can have 0 real roots if the problem is imaginary.

as for having at most 4 real and 1 imaginary, its because the highest exponent is 5, but if you did have 4 real roots then the answer would be 4.