Question

Prove that every polynomial with an odd degree has at least one real root.

Answer 1

it's suppose to use limits I'm guessing it's for a calc test...otherwise I would use the factor theorem...

Answer 2

Limits? I would approach it in a different manner, but this way is also possible.

Answer 3

Suppose you have an odd-degree polynomial \[a_{2k+1}x^{2k+1}+...+a_1x+a_0.\]Now, take the limit of this as \(x\) approaches positive and negative infinity. What do you get?

Answer 4

well you can pick a value where it's below \(0\) and one where it's above so the intermediate value theorem ensures it hits \(0\) in between

Answer 5

thank you guys :-)