SOLUTION: Is it possible for a cubic polynomial function to have no real zeros? Please mention any properties of polynomials in the explanation. Thank you, Lucy

Question 113930: Is it possible for a cubic polynomial function to have no real zeros? Please mention any properties of polynomials in the explanation.
Thank you,
Lucy
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
No it is not possible for a cubic polynomial function to have no real zeros. Let's examine the end behavior of the graph.

If you let x be a really large number, for instance one million, and then cube it, you get an even larger number. So this number f(x) will be positive. If you let x be a really small number, for instance negative one million, and then cube it, you get an even smaller number. So this f(x) value is negative. So this means we will always have some positive y values and some negative y values. Since this graph is continuous, in between these values there must be at least one real zero (ie the graph must cross the x-axis at least once to go from positive to negative and vice versa). So this shows that any cubic polynomial (actually any polynomial of odd degree) will have at least one real zero.

Does this make sense?
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