This week we completed chapter one of polynomials and continued with factoring them in chapter 2.
To factor a polynomial, you first need to find common factors, and find patterns within them.
Some can be solved by finding their specific factors, within the coefficients. You can use their common factors to divide every term of the polynomial, but you need to show it being multiplied by the same number, otherwise it would not become the same answer if flipped around.
But even after that, the factorization has not been complete. To know that you have completed a question, the variables need to have no exponents.
You can find the complete answer by finding a pattern. In the trinomial , it can be simplified to . This is because there are two x’s, making , the middle term comes from adding the 2 constants in the simplified polynomial, and the last one comes from multiplying them.
You can find numbers that follow this pattern in a polynomial quite quickly.
ex. (continuing first example)
This week we began our unit on polynomials. I learned how to group like terms and find specific degrees, coefficients, and kinds of polynomials (monomial, binomial, etc.) One of the things I learned on top of this was how to multiply polynomials using various methods. I will be showing how to use area cubes/squares and FOIL/claw method.
To use the area square method, you can take the specific terms and lay them outside of a square. The square is divided into 4 smaller squares, and outside of the squares lie the polynomials terms. One on the left side of the square, and one on the top. You put one piece of the term over one smaller square, and another over the other.
The squares align so that you can do the multiplication step by step, the two in the corner, both corners, top and bottom corners, etc.
Once you have all of the terms from the multiplication, you will most likely have like terms. You would need to group these like terms together to find the final answer.
The FOIL method stands for the order in which you multiply the numbers. F – first, O – outside, I – inside, L – last.
Just like the area square method, you multiply them, step by step to get the answer. But once again, you may have like terms that need to be grouped to find the final answer.
The FOIL method can also be called the claw method because of the way the lines look. The lined are made to visualize which terms multiply with which.
This week was short, and on top of that, I missed a day because I was at a field trip for my science honours class. But even though it was a short week, I still learned more about trigonometry, specifically word problems and how to use them to find angles and side lengths, something we’ve been learning over the whole unit.
To find the angle of a triangle, you can use two side lengths for the equation (sin/cos/tan) xº , for example: sin xº . With that equation (using the example for the following), you find x by isolating it as in xº , and then you will have the value of xº.
To find a side length, you would use one side tenth, and an angle for the equation (sin/cos/tan) xº , For example: sin 31º . With that equation (using the example for the following), you find n by isolating it, in this case 15 sin 31º . If it were sin 31º , you would use the equation sin 31º = 15 meaning you would have to divide both sin 31º (cancelling it out) and 15 to find n.
I made a simple word problem to find the side length of a triangle based on the height of a person and the suns angle to find the length of the persons shadow:
This week we began our unit on trigonometry. One of the things I learned was SOH CAH TOA, abbreviations that help you to memorize how to find side lengths using angles and equations.
Each set of abbreviations begins with a letter that describes the angle equation to use on your calculator (S=sin C=cos T=tan). Depending on the angle of the triangle you are using and what side lengths are given to you, you can choose the correct angle to use.
Each triangle has 3 sides, and when a base angle is given to you, these sides are given names. The longest of them is the hypotenuse, and the side that the base angle sits on is called the adjacent side, the final one, the opposite side, sits on the opposite of the angle.
The other two letters in each abbreviation shows you what sides to use and in what order (OH=opposite/hypotenuse AH=adjacent/hypotenuse OA=opposite/adjacent. \
Even if you only have one side of the triangle and the angles (90 & other), you can find the side length you are looking for.
This week I learned how negative exponents work and how they effect their base’s. Negative exponents unlike positive exponents don’t increase the numbers size. Normal exponents multiply a number over and over (ex. ) and negative exponents turn the number into a fraction (ex ).
You can turn it into a normal number by following the next steps. I will use for this example.
First, if the exponent is negative, then you turn it into a fraction of .
Then you put the denominator on the bottom, therefor making the exponent positive.
Alternatively, I learned that if the negative was originally on the bottom, you would then move it to the top.
Then you find out what the product of the power is and put that underneath a 1, in this case
I learned that if it were for example , then only the and its exponent would be moved to the denominator as in
If it were , then both would move to the bottom.
if it were would equal because you multiply the exponent.
This Infographic shows the 15 largest moons in our solar system.
I chose to do a drawing/infographic because I love to draw and thought that this project would be a great opportunity to use these skills. I chose to make it based around space because I have always been fascinated by it. I decided to base it on sizes of objects in the solar system, specifically moons. The data table I found gave information on their radius’ (I turned it into diameter in the infographic) and the planet they orbit (+Pluto). I found the project to be a creative and fun way to take data and make it more interesting.
This week, I learned how to find prime factors using prime factorization. Prime factors can be used to find all the numbers factors (not just prime), if it is a perfect square or cube, and even its common multiples with other numbers.
You can find prime factors by following these steps (picture below), I will use the number 96 as an example.
Start by finding the lowest number that it can divide into and put it beside the number, in this case, the 2 is put beside the 96.
Under it, write the quotient of the number (in this case 48), and repeat what you just did. In this case it will divide into 2 three more times, so we will skip them and jump to when it can’t divide into 2.
Now it has become the number 3, which can’t be divided into 2 as a whole number. Plus, it is also a prime number, so it can only be divided by itself and one, so I will put 3 beside it, to turn it into 1.
After you reach 1, you stop.
Then you have all of its prime factors. You can multiply any and all of these together to find all of its factors. Because there are 5 2’s, you can simplify it by turning it into