Remainder Theorem and The Rational Roots Theorem

Today in class, we went over one more time how to do synthetic division and how to find the roots of a function using the Remainder Theorem.

After that, we learned how to find the missing coefficient in a polynomial if we already know what the remainder is. Here's how you would do this:

Let's say you have the question:


The first thing you want to do is write it out in a way that is easy to understand:


Then you plug the root of the denominator into the function:


Since we already know the remainder we can rewrite it this way:


Now all we do is isolate K with a little algebra, and solve it:


And there you have it!

Near the end of the class we managed to quickly learn the Rational Roots Theorem. This theorem allows us to find any rational roots of a polynomial function. Here's an example:

So you're given the equation:


The first thing to do is the find all the possible positive and negative factors of the constant term:


Now we find all the positive factors of the leading coefficient:


We then list all the possible rational roots, eliminating any duplicates:


We can then test out these roots by using synthetic division and the factor theorem to turn the function into a quadratic (Remember: If the remainder is 0, then it is a root):


This then gives you:


Now you just factor the equation and find the roots:


And there you have it! That's about all we did for today, tomorrow's scribe will be...Niwatori-san