Math 10 Week # 11

For week 11 just before the midterm, my class and I were practicing and reviewing the units that we have covered since the beginning of the semester. For this blog post, we were instructed to create a post that relates many of these units together. For this post, I am going to explain the similarities between numbers, and polynomials which is factoring. To review, factors is when you break down a number into two or more numbers that when multiplied together create our original number. Factoring is the process of finding these numbers from our given number. I will break it down how factoring relates to all of these subjects.

For our first unit on Numbers, we learned or reviewed about the different ways we can manipulate numbers, and find what makes them up (their prime factors). In this unit, we used prime factorization to find the prime factors of numbers to find the Greatest Common Factor (GCF) and Lowest Common Multiple (LCM) between numerous numbers. For this example we are going to use the numbers 150 and 420. First we are going to use prime factorization to find the prime factors of both numbers and in this case I used the factoring table (I explain the different ways of doing prime factorization in my Week 1 blog post).  Next to find GCF, we need to multiply prime factors that both numbers share and only that. In this case, they both share 2, 3, and 5 so are going to multiply them together and we end up with 30 as our GCF. To find the LCM we have already found the prime factors that numbers share so now we are going to continue with that. This time, instead of eliminating every other prime factor that the numbers don’t share, we are also going to multiply that with our GCF. In this case, we are multiplying 2 • [2 • 3 • 5 (GCF)] • 5 • 7 and we end up with 2100 as our LCM.

Next I am going to explain how we use prime factorization in polynomials and algebraic expressions. In this unit, we go by a rule called Can Divers Pee Easily Under water. This stands for Common (is there anything in common between terms in a polynomial), Difference of squares (is the polynomial a difference of squares), Pattern (does the polynomial fit the pattern for easy or ugly polynomials), Easy (is it an easy polynomial), and Ugly (is it an ugly polynomial. This sort of rule is similar to BEDMAS as they go in order to complete the expression. So in this unit, we factor polynomials so we can write them as their factors and in this case we will use prime factorization to find the GCF of terms if they have anything in Common. This makes it easier to factor the rest of the polynomial. For this example, we are going to use the expression 8 – 32. At first we begin if the two terms have anything in common or have a GCF. In this case, they do as you can divide 8 from both terms. Below the expression, you write 8 left to the first bracket (because we need to multiply 8 in order to get our answer with the factors) and we write the rest of the equation that has been divided by 8; 8( – 4). Next we check if the expression is a difference of squares. For it to be a difference of squares, all of the numbers and variables have to be squares or be able to be square rooted evenly or this process wont work. The polynomial also needs to be a binomial, if not, this process wont work. And lastly, the leading term needs to be subtracting the second term because we are finding the difference between the terms or subtracting one from another. Our polynomial has to fit under all of these rules in order for it to be a difference of squares and for our expression it follows all of them. Our last step is to factor our expression using the difference of squares and we end with 8(x + 2)(x – 2).