Factoring Polynomials Part 3:

SOAP

The last method for factoring polynomials is SOAP. SOAP stands for Same Opposite Always Positive. The requirement for using SOAP when factoring is that there have to be two terms and there would need to be x^3. You would need to find the cube root of both terms and would use SOAP in order to factor the polynomial. For instance, the polynomial 27x^3 - 64. When using SOAP after you find the cube root of both terms you would need to put them in the form of (a  b) (a^2  ab b^2). When finding the cube root of 27x^3 it would give you 3x which would be a. Then you would need to find the cube root of 64 and it would give you 4 which is b. Then you would take the formula and put them where they belong. So, it would be ( 3x   4) ((3x)^2    3x(4)   4^2). Then you would use SOAP to fill in the sign that would fit in between the terms. 

            S             O     AP

                (3x-4)  ((3x)^2 +  3x(4) +  4^2)

It would be a minus sign for (3x-4) because the regular problem is a subtraction problem so it would stay the same. S represents stays the same. It would be a plus sign because addition is the opposite of subtraction. O represents the opposite. Then for the last sign, it would be a plus sign because AP would stand for always positive so a plus sign would be added in. 

After you are done factoring then you would only need to simplify it. So, the final answer would be (3x - 4) (9x^2 + 12x + 16)

Those are all the 6 methods to factoring but there might be times when you won’t only use one but would have to use other methods to get to the answer. For instance, there could be times when a problem would use GCF followed by AC or there might be a problem that would use GCF followed by SOAP or DOPS followed by still DOPS. 

Example:

An example would be 16x^3 - 4x^2 - 36x + 9. There are four terms which means that you would need to use grouping so you would group the first 2 terms and then the second last two terms. Then it would be (16x^3 - 4x^2) and (-36x + 9). Then you would need to find the GCF for both of them. In (16x^3 - 4x^2) the GCF is 4x^2 so you would divide 16^3 - 4x^2 by 4x^2 and it would be 4x^2(x-1). As for  (-36x + 9) the GCF is -9 so you would need to divide -36x + 9 by -9 and it would be -9(x-1). They both have something in common and that is (x-1) so (x-1) would be factored out. It would then be (4x^2 - 9) (x-1). But it’s all simplified because (4x^2 - 9) can still be simplified further. You would need to use DOPS next because there are two terms and they are perfect squares and there is a minus sign in between the two terms. 4x^2 is a perfect square because 2x times 2x equals 4x^2 and 9 is a perfect square because 3 times 3 equals 9. (x-1) wouldn’t need to be simplified because it’s already completely simplified. So, the final answer would be (2x+3)(2x-3)(x-1).