how come quartics cant have 3 odd x-intercepts??

Question

or can they? it was a question on my functions test that i didnt get...

explain please :)

Answer 1

I'm not sure what your teacher means by an "odd" intercept -- if she just means an intercept at an odd number, that's clearly false. More likely she means something like an intercept with odd multiplicity, and that's true. The reason is that a quartic has four roots, counted with multiplicity, so the sum of the multiplicities of the distinct roots must be 4 -- an even number. If there were exactly three roots with odd multiplicities, then the sum of their multiplicities would be the sum of a list with exactly three odd numbers, and thus odd, and thus not equal to four -- a contradiction.