ex:
![](http://3.bp.blogspot.com/-ZTPRDeipvIo/UHIV-doHiwI/AAAAAAAAAA8/y_Bpz6IPxK0/s1600/CodeCogsEqn-2.gif)
![](http://2.bp.blogspot.com/-4UrE5w3MDSs/UHIWneAKRDI/AAAAAAAAABE/yf6jHVV4ync/s1600/CodeCogsEqn-3.gif)
However, the answer isn't just "x= -2", because there were two of them (a repeated zero). To compensate for that, we say that the solution is -2, with a multiplicity of 2 (because it was there twice).
A factor of the expression below yields a repeated zero x=a of multiplicity k.
- if k is odd, the graph crosses the x-axis at (a , 0).
- if k is even, the graph only touches the x-axis at (a , 0).
So, in the previous example, the graph would touch the x-axis at x=-2 because the solution had an even multiplicity.
Things to remember about zeroes:
- a polynomial with degree n has at most n zeroes, including the multiplicities
- some functions do not ever cross the x-axis because their solutions are imaginary
If you need further explanation on finding the zeroes and determining their multiplicity, this video is really helpful.
More helpful links:
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