Factoring Polynomials Part 1:
Factoring Polynomials is breaking up the polynomial into simpler terms meaning until it can’t be simplified any more. There might be some polynomials that sometimes can not be factored in because they are rational numbers. But when factoring polynomials there are 6 methods that can be used to simplify the polynomial. The 6 methods of factoring are GCF, DOPS, AM, AC, Grouping and SOAP.
GCF stands for the greatest common factor. When using GCF, you just have to find the greatest common factor in the polynomial and divide all terms by a term. For instance, in the problem 6m +15, you have to find the greatest common factor which in this case is 3 and you divide 6m by 3 and 15 by 3 and you get 3(2m + 5). Another example would be 6x^2 + 9x and in this problem, the greatest common factor would be 3x because they both can be divided by 3x and they both have an x. So, the answer would be 3x(2x+3) and when you distribute the 3x to 2x+3 you would get 6x^2+9x which shows that it’s the correct answer.
DOPS stands for the difference of perfect squares. The requirement for using DOPS is that there have to be two terms and there has to be a minus sign in between perfect squares. A perfect square is a number that is created by multiplying two equal integers by each other. Examples of perfect squares are 49, 25 and so on. An example of a factoring polynomial using DOPS is x^2 - 49 and there are two terms, 49 and x^2 are perfect squares and there is a minus sign in between the two perfect squares which means that DOPS can be used. X^2 is a perfect square because x times x equals x^2, 49 is also a perfect square because 7 times 7 equals 49. So, it would be simplified and the answer would be (x+7) (x-7). In DOPS, it would always split in half and there would always be a plus and minus sign in the final answer.
AM stands for Addition and Multiplication. The requirement for using AM when factoring polynomials is that there have to be 3 terms, the first term has to equal to 1 and can not be greater than 1 or less than 1. It has to be exactly 1 meaning that it can be x^2 because there is an invisible 1 in front of the x^2. When using AM, you have to find two numbers that add to the middle term and multiply to the last term. An example of a polynomial that requires using AM is x^2 + 11x +18 and in this polynomial, there are all the required requirements which is that there are three terms and the first term equals 1. To solve this polynomial, we have to find two numbers that add up to 11 and multiply to 18 and those two numbers are 2 and 9, when finding the two numbers you have to check if the two numbers are negative or positive and in this case both numbers that add up to 11 and multiply to 18 are both positive. Since both numbers are positive you would use the plus sign but if the numbers were not positive and were negative then you would use the minus sign. So, the final answer would be (x+2)(x+9).