"Thank God for Control + Z" - Mrs. Bjornson Hey everyone, Felix here to follow up with part two of the removing common factors lesson. In part one, we learned how to remove monomials from an polynomial. In this part of the lesson, we learned how to factor binomials from binomials as well as how to factor four-term polynomials. So when you see a question like x (x + 1) + 3 (x + 1) the first thing you want to do is imagine that x + 1 is equal to y so that the original equation can be rewritten as xy + 3y. Then using what we learned last time we can factor it into the equation y (x + 3). Finally recall that y = x + 1 and replace the y in the last equation with x + 1 so that you get (x + 1) (x + 3), the fully factored form of the original equation. Another way to factor x (x + 1) + 3 (x + 1) is to look over the terms in the polynomial and ask yourself are there any common multiples between them and if there are highlight them. In this equation, there is a common multiple of x + 1 between the terms, so you would highlight it. At this stage your equation would look like this, x (x + 1) + 3 (x + 1). Finally, you would move the common multiple to the front of the equation and put brackets around anything the remained in the terms. If you did that with this equation it would look something like this, (x + 1) (x + 3). Finally, it's time to four-term polynomials. When dealing with these tricky four-term polynomials, what you want to do is to split it into two separate parts before combining them back together and then factoring them one more time. So when you see something like x^3 + x^2 + 3x + 3, the first thing you would do is turn it into two separate binomials, x^3 + x^2 and 3x + 3. Then at this point you would take out the GCF from both binomials, so that you get x^2 (x + 1) and 3 (x + 1). Finally, recombine our two binomials so that you get the statement, x^2 (x + 1) + 3 (x + 1) and notice that this resembles the type of question that we did at the beginning of this post. So we follow those steps and get (x + 1) (x^2 + 3) as our final answer.
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Okay, “lets get this bread!” Getting into factoring which is probably the most important skill to have for math 11. You need to understand Term=types of monomials. When looking at variables attached to terms remember that even if it doesn’t have one, that it actually has a hidden one. Example: just a plain 10 still has a hidden x to the power of 0 which becomes 10 to the power of 1. It is an important skill to remember for equations. Polynomials is a bunch of monomials together by adding or subtracting. Also a monomial -despite Mrs Bjornson’s contempt about it- is still classified as a polynomial. When completing equations, you can reorder monomials. It is better and usually called for to start with the highest degree of the variable and keep going down. But remember to take the sign with the terms! Figuring out the degree of polynomial. Degree is the sum of the exponents of the variable in that term. So it is the term with the highest degree. We have examples of this in our notes on page 64. The key thing to remember is again, don’t forget the invisible 1 on some variables. So when asked to find the leading term and leading coefficient, the term to be in front is the term with the highest degree. Also the leading coefficient shouldn’t be negative. Something that will make things easier is combining the like terms. You take the terms with the same variable (make sure they are also to the same power of exponent or else it will not work) and adding them together so your equation isn’t so long and annoying. For final answer all like terms should be combined, and to add them together you add or subtract the coefficients. Honestly I am not sure if I am good at explaining or helping anybody at all, let’s just remember I am trying my best :’). There was a quick hat break during this part of the lesson which lead to a like terms example and I would talk about this, but I got distracted by them, sorry guys. What can be added using algebra and hats a Mrs Bjornson example: hat + hat = (h+h) is a yes Suggest trying algebra tile if you are struggling with Hat + food= (h+f) is a no this concept. Food+food= (f +f) is a yes Here are a couple (hopefully) helpful reminders when multiplying a monomial by another monomial. First multiply the constant factors, then variable factors. You must follow exponent laws: PMA. Steps on what to do:
Yikes I am bad at this but I am almost done, so I am just gonna power through. The time to use the rainbow rule is applied when multiplying a polynomial by a binomial. You must use that rule to get rid of the brackets. If you aren’t doing that, then you are most likely doing something wrong, so keep that in mind. Okay so thanks for reading this if you did, and being patient.I hope this was a little helpful. (Lanie) Hello. Today we talked about removing common factors from polynomials, but we didn't make it through the whole lesson so this is only part 1, dealing with removing common factors from monomials. First of all, factoring is the opposite of expanding (multiplying). For example, 5x + 10 = 5(x + 2) is factoring, whereas 5(x + 2) = 5x + 10 is expanding. Removing a common factor is the most basic form of factoring. Common Factors are factors that are common among the terms in polynomials. To find them you can use a factor tree : Once you have found the GCF, put everything else in brackets. For example, 15c^2 - 45c = 15c (c - 3), the GCF of 15c^2 and 45c is 15c. Next, divide 15c^2 - 42c by 15c. There is a "c" left over and 45 ÷ 15 = 3. In summary, the steps you should be taking to remove common factors are these : 1) Determine GCF 2) = GCF ( ___ ) ← for monomials = GCF ( ___ + ___ ) ← for binomials = GCF ( ___ + ___ + ___ ) ← for trinomials etc. etc. This was only part one of our lesson, so there really isn't too much left to cover. If you have any questions, this video should explain them well. Other than that, have fun removing common factors :) - Lucy Hello my friends! Today we learned all about the glories of multiplying polynomials. Fear not for this unit is mostly review from grade 9. So we should all be fine. Right? The first thing to remember about multiplying polynomials is that we are using the distributive property. I will Now explain a couple ways to multiply polynomials. 1. The first one I will discuss is the Tic- Tac- Toe Method. This method uses a little tic-tac-toe graph to multiply all the terms together and find the like terms. This method is honestly very similar to the Punnet Square which we did in science class. I have provided a photo of an equation being simplified using the Tic-Tac-Toe Method. 2. Next I will talk about the FOIL Method. This is my method of preference as I find it the most simple. The letters in FOIL stand for First, Outside, Inside and Last. These words are for helping you remember which terms to multiply so that none are left out. Here is a video of a person teaching how to multiply polynomials using the FOIL method. He goes VERY slowly so if you feel that math class goes too fast you might enjoy the video. The last thing to remember when multiplying polynomials is the shortcuts about squaring a binomial. The phrase which Ms. Bjornson told us was "Square the first, multiply the terms and double the answer, square the last". Here is a video of a girl squaring some binomials and she basically does the exact same thing Ms. Bjornson explained to us. This video goes a bit over the top with examples so you probably don't need to watch all of it I'm sorry if this was not the best blog post, I'm pretty bad at explaining things bit I hope the photo and the videos will help any of you guys who are confused.
Nobody missed this class so I doubt many of you will see this post but anyway I hope you have a nice day! - Olivia |
Find your green dot!AuthorsWe are members of Esquimalt High School's Gifted Math 10 class. Most of us like Hi-Chews. Archives
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