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

Using Venn Diagrams in Word Problems

In class today, we went over some questions on Venn diagram sets. You can see the answers to in the slides that Mr. K posted.

After doing a bit of that, we then learned how to apply our knowledge of Venn diagram sets and rules to problems that you would see in real life.

Let's see how we would solve problems that look like this with Venn diagrams:

"In a group of students 12 are taking chemistry, 10 are taking physics, 3 are taking both and 5 are taking neither. How many students are in the group?"

First we would draw our 'universe' and the Venn diagram that goes in it.:
('U' being universe, 'C' being chemistry, and 'P' being physics)
After you have everything set up, it's time to add in some numbers. Let's start with the easiest possible ones first.

Seeing how there are 5 students who aren't taking either chemistry or physics, we can tell right away that they're going to be placed somewhere in the 'universe', outside of the Venn diagrams.

The 3 students who are taking both courses are the next obvious choice. Since they are taking chemistry AND physics, they belong in the intersection, where the 'C' and the 'P' circle meet.

Now we'll figure out how many students are taking each course.

Automatically your brain might think that the number of students taking chemistry is 12 and the number of students taking physics is 10, but neither statement is true.

The actual number of students taking chemistry is 9, and the number of students taking physics is 7. Why? Because we exclude the 3 students that are taking both courses.
We subtract those 3 students from the pool of students taking chemistry (12 - 3 = 9) and physics (10 - 3 = 7).

We do this because in circles 'C' and 'P', we only want the number of students who are ONLY taking either chemistry or physics, excluding the students that are taking both.

Using the information we've gathered, we can determine the total number of students by adding up all the numbers. You should then finish off with this:
And that's how you solve a word problem with a Venn diagram.

We did a couple more examples, which all use pretty much the same procedure.

We did however, do one that involved 3 circles in the Venn diagram instead of 2. It seemed a little confusing, but solving it involved the same procedures as the previous examples. You just had to be very careful with your numbers.

That's about it for my scribe post, thanks for reading. The next scribe will be..camiLLe

Welcome Back Mr. K!

Here are the topics that I'm having the most trouble on

* The Ambiguous cases of triangles.
* Circles
* Radical and Rational Equations