I made about as much progress this week as I have every week since I began Precalculus, i.e. didn’t get too far. Just like every other week, I once again feel like I got a lot of really good work done but it’s taking me longer than I anticipated and longer than I’d like it to. I finished the unit Complex Numbers on Tuesday (sort of) and spent the next four days working through the unit Polynomials but only got 40% of the way through it. I breezed through the first section of the unit but was then introduced to a new concept I’d never learned before, the binomial theorem, which took me three days to get a good understanding of. Nonetheless, I’m still happy with what I was able to learnt his week and would even say it was one of the most productive weeks to date. 😊

(Also, I just learned that the command keys to access apple’s emoji library which is Control+Command+Space. Get ready for all the emojis! 👍🏻)

I managed to pass the Complex Numbers unit test on my first try Tuesday morning. Going through each of the questions, I felt the usual amount of nervousness and doubt and when submitting my answers but at the same time felt like I truly understood the questions and understood how and when to apply the proper formulas. I scored 100% on the test, however my overall score for the unit is still at 99% because the unit test didn’t offer me a question I needed to bump up a specific section’s score to 100%. I decided not to redo the test since the questions I get on it are out of my control and mostly because I wanted to move on. Plus, I feel confident with my understanding of the section that’s not at 100%, Polar Form of Complex Numbers, so I’m not too concerned.

Getting into the unit Polynomials on Wednesday, I was happy that I remembered the definitions and terminology that came up. A few key terms that are used when working with polynomials are:

• Polynomial
  • “Poly“ means “many”, i.e. “poly-nomial” = “many-terms”
  • Binomial = 2 terms (ex. (x^2 + 1))
  • Trinomial = 3 terms (ex. (2x^2 + x – 4))
  • A polynomial is “an expression that has a bunch of variables or constant terms in them that are raised to non-zero [and non-fractional] exponents.” – Sal
    • Ex. (x^(-1/2) + 1) = (1/√x + 1) is NOT a polynomial since it’s raised to a negative, fractional exponent. 
    • I believe this means is there’s a root of any kind in the expression (i.e. squared-root, cubed-root, etc.) it’s not a polynomial.
      
• Degree
  • Refers to the highest exponent in a polynomial
    • Ex. (4x^3 + 2x^2 + 4) = 3^rd Degree
      • (x^2 + 1) = 2^nd Degree
      • (x^4 + 1) = 4^th Degree
      • (1 + x^10) = 10^th Degree
        
• Coefficient VS Constant VS Variable
  • A coefficient is a “multiplier” that is typically placed at the beginning of a term that includes a variable.
    • Ex. in the term 4x^2, the coefficient would be 4.
  • A constant is a term that is a standalone number whereas a variable uses a letter or symbol to denote a value and may or may-not have a coefficient in front of it.
    • Ex. in the polynomial (4x^3 + 2x^2 + 6) both of the terms 4x^3 and 2x^2 are considered variables whereas the final term, 6, is considered a constant.
      
• Standard Form
  • Lists a polynomial in order of degree from highest to lowest.
    • Ex. taking the polynomial (1 + 4x^4 + x + 2x^2) and changing it standard form would give you (4x^4 + 2x^2 + x + 1).
    • I’m not 100% sure, but I believe if there’s more than one variable in the polynomial it’s good practice to list the variables in alphabetical order but I’m not sure if that’s considered a part of standard form.
      • Ex. (2 + 3yx + 5xy^2) would change to (5xy^2 + 3xy + 2).

On Thursday I was introduced to what’s known as the Binomial Theorem which was something completely new to me and was what I worked on throughout the remainder of the week. This theorem is used to figure out the answer to any binomial raised to any power, (ex. (x + y)^2 or (a + b)^4, etc.). I learned about 3 methods to help find the answer to these types of questions, but the main formula to solve these questions, a.k.a. the binomial theorem, is:

Even having worked on understanding this theorem for three days, I still find it a bit confusing and intimidating when I look at it. The variables in the equation stand for the following:

• a and b are the two variables
• n is the power/exponent on the binomial
  • (Side note: n + 1 will the total number of terms which is helpful to know.)
• k is the term-number
• Σ = “sigma” = “sum of all the terms from k = 0 to n”

Here’s an example of how the formula is used:

When raising a binomial to any power, there is always one more term in the final answer than the value of the power, so in the example above there are five total terms in the final answer since the exponent, n, is 4. The first term begins with _4C_0 (pronounced “four-choose-zero”) since below the sigma symbol you see k = 0 (read: “first term (k) begins at 0). For each successive term you add 1 meaning in the second term k = 1, in the third term k = 2 etc.

Although this formula seems confusing, it eliminates the need to do quite a bit of extra algebra in order to calculate the final answer. Here’s how you would solve (a + b)^4 if you were to use the FOIL method:

After learning about the Binomial Theorem, I then learned about another method to solve these types of questions that uses a magic triangle known as Pascal’s Triangle which isn’t actually magic but seems like it is:

You’re able to figure out the coefficients of binomial being raised to n by beginning at the top row of the triangle and moving down n rows. As you can see from the second photo, if you wanted to figure out the coefficients on a binomial raised to the power of 4, beginning from the top row you move down to the 5^th row where there are 5 numbers, 1, 4, 6, 4, and 1, which are the corresponding coefficients on each term. Except for the 1’s on the sides of the triangle, each number is the sum of the two numbers above it. Each number can be thought of as a node within the triangle and equal the total number of paths you could take going from the top of the triangle down to that number/node.

The following photo explains how to find the exponents that both variables are raised to at every node within the triangle:

(down to the right*)

On this version of Pascal’s Triangle the variables I used were x and y. As stated in the photo, every time you go down a row to the left you add an x and every time you go down a row to the right you add a y. So, for example, going down the triangle 3 times to the left and 2 times to the right gives you 3 x‘s and 2 y‘s which lands you on (10)x^3y^2. It’s worth noting that the exponents on both variables will always add up to n (ex. taking (x + y)^4 you see that the exponents at each node in the fifth row always add up to 4.)

There is one more method I learned to find both the coefficient and exponents on each power when raising a binomial to n. Here are the pages from my notes that explains how this is done:

All in all, it was a pretty difficult but satisfying week. Understanding how to raise binomials to powers of n was tough to wrap my head around but I think I managed it pretty well and now have a pretty good understanding of the three different methods. This coming week I’d like to get through Polynomials (320/800 M.P.) and get started on the following unit Composite Functions (0/700 M.P.). I’m hoping to get through Polynomials by Thursday but, based on the rate I’ve been going lately, I wouldn’t be surprised if that doesn’t happen. It seems like the further I get into KA the longer it takes me to get through units. But on the bright side, at least I figured out how to insert emojis! 😊 😆 😋