Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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Once we have a thorough understanding of polynomials we can look at **rational functions** that are a quotient of two polynomials. These rational functions have certain behaviors, and students are often asked to find their limits, or to graph them. Their graphs can have different characteristics depending on whether the numerator function has degree less than, equal to, or greater than the denominator function.

I want to talk about a very important class of functions called rational functions. A rational function is one that can be written f of x equals p of x over q of x where p of x and q of x are polynomials.

Now, f of x is defined for any number of x unless q of x the denominator equals zero so the domain will be all real numbers except those that make the denominator zero. And the zeros of a rational function will be the zeros of the numerator just as long as they are not also zeros of the denominator, so let's practice using these definitions in an example.

Each of these three is a rational function, polynomial divided by polynomial so p of x over q of x. Now, find the domain and zeros. The domain of this function is going to be all real numbers except where the denominator is zero, so where is the denominator zero? 2x-5=0 when 2x=5, so we divide by 2, x equals five halves so the domain is all real numbers except five halves, all real numbers except five halves, now what are the zeros? For the zeros we look to the numerator. When is the numerator equal to zero? 2x squared minus 5x minus 3. Now this looks like it's factorable so I'm going to try to factor it 2x, x. I need a 3 and a 1 now if I put -3 here and +1 here I'll get x-6x, -5x that works. That means that x equals negative one half and x=3 are both zeros of this function and because neither of those zeros is also a zero of the denominator, these are going to be zeros of my function so the zeros are negative one half, x=3.

Okay let's take a look at this guy, what's the domain? Well first we have to figure out where the where the denominator equals zero, so x squared minus 4x equal 0, I can factor this it equal zero when x is 0 or 4, so the domain will be all real numbers except 0 or 4, all real numbers except 0 or 4, now for the zeros of the function the numbers that make this function 0 we look to the numerator, x squared minus 1 equals zero and that's really easy x squared equals 1, x equals plus or minus 1, so as long as plus or minus 1 are not also zeros of the denominator, these are zeros of my function so the zeros are plus and minus 1.

Finally let's look look at this function, this denominator I can find the zeros by factoring, x cubed minus x squared minus 6x equals 0, so you get x times x squared minus x minus 6 and this can also be factored looks like it's going to be x and x. I need maybe a 2 and a 3 if I go -3+2 I get my minus 6 and I get -3x+2x negative x that works, so the zeros of the denominator are x=0, 3 or -2, so the domain will be all real numbers except those three. Domain all reals except 0, 3 or -2. And then what about the zeros of this function? Let's look at the numerator; x squared minus 4 equals zero means x squared equals 4 so x is plus or minus 2. Now here's a case where one of the zeros of the numerator is also a zero of the denominator now because 2 is a zero of both the numerator and denominator, the function is not going to be defined there so you can't say that the function's value is 0 there, its not a 0 the only 0 will be then be -2 again I'm sorry actually -2 is this is the it's where it's undefined so positive 2. Let me just clarify, the function is not defined at -2 so -2 can't be a zero so it has to be +2 only.