How to Find Real Zeros of Any Polynomial Function 

This was in algebra 2 for like 1 week barely a day lmao

The answer is very simple my friend. Cause in the first case when you divide a polynomial, you use a divisor which is a factor of the polynomial and then multiply and subtract. However when it comes to the Synthetic Division, you do actually multiply it by the inverse of that factor, for that reason you add them. Let me elaborate, 1. For Long Division, we take the divisor as (x-a). 2. For Synthetic Division, we take x=a. Clearly both of them are additive inverse of each other, so that's why we add at one whereas subtract to the other to have them in Equilibrium. It went too long I guess 😅😅😅

What synthetic division does, is replace operations such as subtracting a negative, with adding a positive. It recognizes the self-cancellation of the negative signs, and replaces it with the more intuitive operation of simple addition, and simple multiplication. It makes it so you don't need to guess terms, but sets them up in a more straight-forward method of calculating them

😂😂😂

In that case, your candidates for roots would be: 1/16, 1/8, 1/4, 1/2, 5/16, 5/8, 5/4, 5/2, 1, 2, 4, 5, 10, 20, and 100 And the negatives of all of the above. Usually, you try to use other clues as well, such as Descartes' rule of signs, and the sum and product of the roots, which can be directly determined from the b-term, the final constant term, the a-term, and whether the degree of the polynomial is odd or even. This allows you to narrow down your search for possible roots. Polynomial roots in general, will add to -b/a, and will multiply to k/a for even-ordered polynomials, and multiply to -k/a for odd-ordered polynomials.