Zeroes of Functions and Multiplicity

     The first step to finding the zeroes of a function is to replace y or f(x) with 0. From there, you can use different methods to solve, depending on the degree of the polynomial.  For a quadratic, you can factor it or use the quadratic formula.  If the equation has a degree larger than two, you can use synthetic division, factoring by grouping, etc.
     ex:

   

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:

http://www.purplemath.com/modules/polyends2.htm 

http://primes.utm.edu/glossary/xpage/Zero_.html