Suppose we have a polynomail of degree n:
p(x) = a0 + a1x + a2x^2 + ... + anx^n
where n is an odd number and "an" (the constant) not equal 0. Show that this polynomail has a real root. You can assume that polynomails are continuous.
So it is pretty intuitive that p(x) as x-> inf is positive and as x-> -inf is negative. I just don't know what facts about polynomials to get
a p(a) = negative number and a p(b) = positive number. Then I know you can use the intermediate value theorem.
Say without lose of generality. Define the following sequence . The claim is that . To see that you can write the first term but the second term since it means this limit goes to . Thus there is a so that . Similarly define . And show thus there is a so that . Meaning we can choose an interval large enough so that the polynomial changes from -1 to 1. Which completes the proof.