The Zeroes of Polynomial Functions

Mathew Kuruvilla 8-4

If you want to find the rational roots of a polynomial function, there are two different ways you could use. One is a type of guess and check and the other is basically graphing.

This is the guess and check type way:

The rational root theorem says that, to put it in simplest terms, take the factor of the constant term (P), and divide it by the leading coefficient (Q). (P/Q). These answers, P/Q=…, have to have both + and – and also have to be in simplest terms.

Using this theorem you are able to get all the possible roots for a polynomial function. With these possible roots, you plug in all the values into the function till you get all the correct zeroes of the function.

This is the graphing way:

First, you graph the equation on your calculator, or wherever, and look at the graph. There should be at least one zero of the function that you can identify properly right off the bat.

After that, you use synthetic division with both the zero and the polynomial together.

If the zero you find is true, then you look to where the other zeroes can be found. The number of zeroes that will occur will correspond directly to the highest exponential value in the graph. So, it should be fairly easy to figure out how many more zeroes to look out for. You then look to where the graph intersects or touches the lien and zoom in to get a good view of the zeroes. From there you find the zeroes and again use synthetic division to figure out if the zeroes are correct.

There is a slight varied method you could do though, instead of using graphing, you could also use the quadratic formula, too.

This is another theorem, the complex conjugate root theorem:

This also another theorem, the Fundamental Theorem of Algebra:

These will both eventually be used in the context of finding zeroes of polynomials.

Besides regular problems with polynomials to find zeroes for, there can also be word problems, too. For example:

What you first need to know to solve a problem with boxes like this is that it will always come in a form like this:

H*w*l=X*(width-2X)(length-2X) in which the X is the height.

So, the first thing you do to solve the problem would be to put the all the information into polynomial functions.

You can then graph both equations together and simply find the intersection of both graphs.

This is another type of polynomial problem that may be given:

To solve this problem, you first need to make all the factors of the polynomial equation that needs to be found and if there are any multiplicities, put them under an exponential factor.

Then, since P can be stretched, and you don’t know what the stretch factor is, you have to use a variable in its place and have to find it before moving on. Preferably, use a fro the variable.

After you have found the stretch factor, put the factor in front of the other factors and multiply everything together to get your answer.

THAT’S ALL THERE IS TO KNOW ABOUT THE ZEROES OF POLYNOMIAL FUNCTIONS! GOOD LUCK ON THE FINALS!!!!!