I’m happy to report that this week 1) the world didn’t end and, 2) I managed to get through the two units Polynomial Factorization and Polynomial Division. Considering how distracted I’ve been with everything that’s going on, I’m happy with the progress I made, though part of me does feel like I should have been able to get more done than I did.

First off, a quick update on the corona-virus situation. As of now it’s still growing at an exponential rate. I’ve been tuning into the news less since I feel like things haven’t been changing quite as rapidly as they were last week. The reality of the situation has settled in my mind a bit more and I don’t feel the need to be checking the news as frequently as before. I definitely feel that it’s been better for my mental health not to be checking every single update especially considering there’s nothing I can do about it. People are slowly starting to abide the recommendations to socially distance themselves from others but it’s clear some people are taking it more seriously than other. I think I’m at the end of the spectrum of people that are taking it very seriously and would say I’ve been very good about not going out unless it’s necessary. I’ve also been washing my hands as soon as I get back when I do go out and trying not to directly touch door handles, push elevator buttons, etc. At this point, there’s nothing to do but wait and see what happens (which is one of the hardest things of all of this).

Bringing this back to the work I got through on KA this week, I began the week by finishing the unit Polynomial Factorization. The only thing left to tackle in this unit was a section on geometric sequencing. Going through it, I learned that a sequence is a list of numbers that simply follow a specific pattern. For example, {2, 6, 18, 54, …} would be a sequence with a common ratio of 3 (i.e. each successive term is multiplied by 3). A series on the other hand is just like a sequence but has a finite set of terms which you add together, ex. {2 + 6 + 18 + 54} = 80. Going through this unit and being taught the formula used to calculate a series’ value (see below), I realized it’s helpful to visualize series in the following way:

• {2 + 6 + 18 + 54}
  • = {2 + (2 * 3) + (2 * 3 * 3) + (2 * 3 * 3 * 3)}
  • = {2 + (2 * 3^1) + (2 * 3^2) + (2 * 3^3)}

In order to find the sum of a geometric series, I was first taught the following denotative terms:

• a = First term in the series
• r = Common ratio
• n = The number of terms I’d be adding together, and
• S(n) = The sum of n number of terms
  • Ex. S(3) = The sum of the first 3 terms in a series

Using the above denotation, I’d input values into the following formula:

• S(n) = (a(1 – r^n))/(1 – r)

The part of this formula I found to be the most tricky was taking r to the power of n, especially when n was a fraction. That said, like always, I became better at it the more a went through the questions. I was able to get through the unit test with a 100% score on the first attempt and then moved on to Polynomial Division.

Learning how to divide polynomials was not what I expected and incredibly satisfying once I got the hang of it. I unfortunately can’t think of a good way to demonstrate how to use polynomial long division on here on this blog. The process requires you to first write the equation down in standard long division format. From there, you begin by dividing the first term in the numerator by the first term in the denominator, write that part of the quotient above the second term in the numerator, then multiply both terms of the denominator by the first term of the quotient and subtract those values from the numerator. You then repeat the process until you’ve divided all the terms in the numerator by both terms in the denominator and are left with either 0 or a remainder. (Trying to put it into words, I’ve now just realized how difficult this is to explain…)

The second part of the unit was about the Polynomial Remainder Theorem which took me a while to wrap my head around. It helped me by comparing it to an easier division question:

• Numerator/Denominator = Quotient + Remainder
  • Ex. 25/4 = 6 + R1
• The same thing works for polynomial long division:
  • P(x)/(x – a) = q(x) + R
    • Ex. (3x^2 – 4x + 7)/(x – 1) = (3x – 1) + R6
• You can rewrite both of the above equations as:
  • 25 = (4)(6) + R1, and
  • P(x) = (x – a)(q(x)) + R
    • Ex. (3x^2 – 4x + 7) = (x – 1)(3x – 1) + R6
• The Polynomial Remainder Theorem states that if you make the denominator = 0, the quotient then becomes 0 since you’re multiplying them together and, by doing that to both sides to keep them equal, you’re left with the remainder. Ex.:
  • 25 = (4)(6) + 1
    • 24 + 1 = (4)(6) + 1
    • If you multiple 4 (the denominator) by 0 then you get:
    • 24 + 1 = (4 * 0)(6) + 1
    • 1 = 1
• The same thing works with polynomial division and, therefore, if you figure out what value is needed for x to make the denominator = 0 and input that value into the initial equation, you can tell if and what the remainder would be.
  • (3x^2 – 4x + 7) = (x – 1)(3x – 1) + R6
    • To make the denominator ((x -1)) equal 0, x needs to equal 1. Putting that into both sides of the equation gives you:
    • (3(1)^2 – 4(1) + 1) + 6 = (1 – 1)(3x – 1) + R6
    • (3(1)^2 – 4(1) + 1) + 6 = (0)(3x – 1) + R6
    • 6 = 6
• This makes it easier to tell if a denominator factors perfectly into a polynomial (i.e. there’s no remainder) and also makes it easier to simply find the remainder if that’s all you want to know.

It took me 3 attempts to get through the unit test. The questions I got wrong in the first 2 attempts were due to simple mistakes I made by rushing through the long division process. I made most of my mistakes when subtracting negative numbers in my head which I think is a good thing for me to be aware of so I can slow down when working through those parts.

It’s hard to believe I’m now going into Week 30. I’d like to get through the next 2 units this week which are Polynomial Graphs (0/500 M.P.) and Rational Exponents (150/1000 M.P.). Considering that I’m off work and have so much extra free time, I should be able to get it done. The issue is that, with everything that’s going on with the corona virus, I find I’m still very distracted. For the most it’s been getting easier to focus but recently I’ve been beginning to worry about the safety of people that I know personally which is making it very difficult to focus. Considering this virus is growing at an exponential rate, I’m not sure working through a unit on exponents is going to make it any easier…