Roots/Zeros/Solutions- ALL the same thing! Solutions to an equation when it equals 0.
The relationship between roots and factors, is that when something is in its factored form, it is easiest to find all of the roots. For example:
(x+3)(x-2)(3x+5) has roots x=-3, 2, -5/3
We can also use roots to determine what the factors must be. For example:
Given roots x= -2, 1, and 3/4, the original polynomial must be (x+2)(x-1)(4x-3).
If an equation factors to something like (x+2)^2 (“x plus two squared”) , then x=-2 is called a double root, because it is a solution to the two factors, (x+2) and (x+2).
Rational Root Theorem
Look for you options of p and q, to find your options of p/q, which are the potential roots of a given polynomial. Remember, your options for p are +- all the factors of the last term (the constant), and your options for q are +- all the factors of the leading coefficient.
When using synthetic division, be careful about what “a” represents. We use synthetic division for different things.
1.) Find a remainder- solve for f(a): When you do synthetic division of some polynomial and we put ‘a’ in that little top left box, whatever remainder you end up with is the solution to the polynomial at x=a.
2.) Divide a polynomial f(x) by x-a: When you do synthetic division of some polynomial and we put ‘a’ in that little top left box, you are dividing the polynomial by x-a. Be careful: for example if you want to divide by x+3, you actually are using a=-3. The bottom line of the synthetic division problem, represents the coefficients of the polynomial of your answer. Remember, this answer starts with a variable of degree one less than the polynomial you started with! When you start with x^4, your answer is going to be of degree 3, for example x^3. The last number on this bottom line is the remainder
3.) Determine if x-a is a factor of f(x): Just like above, you interpret your answer the same way, you are just looking for a remainder of 0. A polynomial is only a factor of something else if it does not leave a remainder after division. Also don’t forget that your answer after synthetic division is also a factor of the original polynomial. This new polynomial may also be factorable, so don’t forget to check that!