Back to Basics: Polynomial Functions (featuring factoring and long division)

Good [insert time of day] to you all!

For those wondering, my name is Wandaly. I will attempt to fulfill several goals in my scribe. First of all, I will try to summarize today’s class. Hopefully, I will be able to help you understand the concepts we’ve tackled a little bit better and clear up any confusion. Lastly, I’ll try to make this scribe as entertaining as possible. 

Factoring is the process of breaking down an equation or a statement or a term. There are different methods:

Using the Greatest Common Factor: the method is self explanatory. You look at the possible factors of the numbers and take the largest one.

Factoring by grouping: you group the terms together in a polynomial and factor our the common factor from each group. From there, you will be able to see that both of the groups will have the same factor. You can lump the ones not in a group together. 

You may have encountered factoring the difference of perfect squares in grade 11. This one may be the easiest to factor out of all. Simply look at the two terms and square root each one. Write the terms with opposing signs. 

But wait!

We may overlook simple things. At first glance this question may seem like a trick question:

If we stick to what we've learned, we need to take the square root of x squared and square root of 10 as well. It isn't a nice whole number, but it's still a number.

This is the same concept with factoring difference of cubes. The same can also be said for factoring sum of cubes. The important thing to remember when doing questions like these would be to watch out for the signs. 

The formula for difference of cubes would be:

Meanwhile, the formula for sum of cubes would be:

Notice that the only thing that varied between these two formulas was the sign.

We factored perfect square trinomials as well. Each one can be factored as a square of a binomial. A quick way to do it would be to look at the square root of the first and last terms. 

Our work can be reduced if we are able to remember these tips and tricks. Using the form x squared + bx + c: 

1.) If the c is positive, then u and v have the same sign. They are both positive if b is positive. They are both negative if b is negative.

2.) If c is negative, then u and v have opposite signs.

3.) The variable b is the sum of u and v.

4.) The product of u and v is equal to the product of ac.

During the afternoon class, I was pleasantly surprised when I saw the papers with long division questions. I’m sure that’s not a sentence you read often. However, it reminded me of my elementary days in the Philippines where my teachers used to make us do long divisions the whole class. Calculators were foreign objects to us back then and I’m a bit thankful now.

Personal anecdote aside, I can promise you that polynomial long division uses the same methods as normal long division you have encountered before. It may look intimidating with the powers and letters thrown in, but if we’ll navigate it in baby steps, we’ll be able to solve it in no time. 

First step:

1.) Arrange all the terms in descending order of power.

2.) Ask yourself: how many times does the first term in the divisor go in the first term of the dividend?

3.) Multiply the quotient by the divisor.

4.) Subtract.

5.) Take the difference, check how many times does it go to to the divisor.

6.) Keep repeating the steps until you have divided everything or a remainder appears. A reminder is a degree lower than the divisor.

Perhaps if you're a visual learner, you might learn better if we look at an example.

P.S, If all of what I’ve mentioned still made as much sense as the Queen of England driving her Range Rover, I’ll hand the keys over to KhanAcademy. 

Don’t worry – it will eventually make sense. Look it’s Queen Elizabeth driving her Range Rover! It happens.