For the question: How many roots does the function f(x)=x^9 have?
Would it be 9 or 1, since the degree of the function states it would have 9 but there is only 1 distinct root. I am confused because on sparknotes it said that if the question asked for the roots only and not distinct roots then it would be 9 but I have always learnt it as 1. Thanks
@bobby21 I would think “roots” asks for all roots (including multiple roots). The fundamental theorem of algebra states that an nth degree polynomial has exactly n complex roots, including multiple roots.
I see, so even if there is only one x-int on the graph, there would still be 9 roots right? Thank you
@bobby21 Yes - nine roots, but one distinct root.
From Wikipedia [article](https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra): The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots.
Remember this only gives you real roots. The polynomial f(x) = x^2 + 1 has no x-intercept on the xy-plane, but two distinct complex roots.
A while back, actually, there was a question on the SAT II Math 2 which included a diagram of a parabola with its vertex on the x-axis. It asked how many roots the corresponding quadratic function had. The correct answer was 2.
I see thank you guys
Just recalled a somewhat related question: for what values of k does the graph y=x2+k intersect the x-axis?
Since a point of tangency is also an intersection point, the right answer choice is k<=0.
@gcf101 Do you know if Math II has ever asked a question like this?
For what real values of k does the graph y = x^3 + kx^2 + 2016x + 1 intersect the x-axis? (clearly I made up 2016)
Because the graph has odd degree, it tends to -∞ and +∞ and must cross the x-axis. I wouldn’t be too surprised if they did…
I don’t recall seeing questions like that, but I think it would be a good candidate.
I’ll know better tomorrow. 
@MITer94 There was nothing like that on todays’ either math 1 or 2. There was a cool question on minimizing a distance between two points on the axes, otherwise nothing to write home about: zeroes/domains/limits of quadratic/rational functions, a period of a piecewise function, extrema of a simple trig function, etc… One staple on both levels: f(g(x)) = q(x), g and q linear, blah-blah…
Something annoying: boxes and whiskers plot on Level 1 (this is AP stat, right?) and least squares regressions on both maths.
@gcf101 hmm interesting…
I don’t remember the last time I saw a box-and-whisker plot anywhere besides a textbook or a standardized test question. However I do recall least squares regression on Math II.