This week in Pre calc 11 we learned how to factor polynomials. One of the ways we can factor a polynomial is through using substitution.
Lets suppose you have something like –
Step 1: for substitution you can just substitute a variable of your choice to equal the terms in the brackets so that it can become factorable –
Step 2: once it becomes factorable you can start factoring, in this case I’m going to be using the box method –
Step 3: when we substituted we just made this polynomial factorable so that we can factor it and make it easier but it doesn’t mean that it’s the final answer. Now that we have the two multiples we can add them to the original (x-6) that we substituted a for –
This week in Precalc 11 we learned how to factor an “ugly trinomial” which is a trinomial that has a coefficient in front of the . This is when we can’t use our factoring 1 – one thing in common, 2 – 2 terms (difference of squares), 3 – 3 terms pattern (product & sum). Factoring a trinomial is actually kind of easy once you have figured out the way you like to solve them. There’s two ways, the first way is just guessing and plugging in numbers which can be difficult because you are guessing until you find the right number and that can take forever. The second way is to use a visual piece that’s called the box method. I personally like it better this way because it’s more straight forward and efficient. So today I will be showing how to factor an ugly trinomial using the box method correctly.
This is the ugly trinomial –
Step 1 create a square and plug in the first and last terms in the trinomial *the first term goes in the top left hand corner and the last term goes in the bottom right corner –
Step 2 fill in the empty spots in the box. Do this by multiplying the first and last terms together. Then you find the multiples of the number until you have hit the two that add or subtract to the term in the middle which in this case is our -5mn. Once you have found your two multiples then you plug in those numbers into the box *it doesn’t matter which spot you place it in so long as it’s an empty one *if the leading coefficient sign has more positives than the negatives than the bigger number in the multiples is going to be positive –
Step 3 find what’s common and there you go. You find whats common across and then up and down *whatever sign the term is on the outside is going to determine the sign of the term that’s common –
To check if you did it right just use foil and see if you get the trinomial you had to start with –
Here’s another example –
A question that I came across this week that stumped me was:
Now for this question I distributed and used foil properly and when I was solving it, it was too difficult because this is what I was trying to break down all at once. Which…led to me making silly mistakes. A tip that I received from my teacher was to do them seperately and then combine them at the end once they are easier to manage.
So I drew a line in the middle so that I know not to combine them and redid the question, which then gave me the correct answer. Now you might be wondering what I’m talking about because even though I did them seperately, once I combined them together the question looked just as long as it was before. Well the size of it didn’t change but what did change is that we now have like terms that we can combine which takes only a few seconds as oppose to a few minutes trying to find where you made a mistake if you did it the non organized way.
This week in class we started unit 2 called Radical Operations and Equations , and the first thing we discussed was how to simplify radicals. We went from an entire radical to a mixed which we already knew how to do from the previous unit so I felt pretty good about it, and then we added a variable with an exponent to the radical like When I took a look at this I understood how to turn it into a mixed radical except I had no idea what to do with the variable with the exponent. So here’s the way that I deal with that variable and some tips that will make it easier for you to remember how to deal with that variable when it pops up in more complicated questions.
1st – Find the prime factorization, when you have a variable with an exponent you also find its prime factorization. * Realize that is a squared root therefor you are looking for pairs when finding the prime factorization. What I do right away is highlight my pairs right away so that way I won’t miss them.
2nd – once the prime factorization is found you will take the pairs and square root them and put them on the left side of the radical symbol taking it “out” or “removing it from the radicand area and then taking whats not highlighted multiplying them together and putting it on the inside of the radical symbol making it the radicand.