Basically everywhere in math, finding the roots of a polynomial is a very important part of algebra. That's why I have created a beautiful method of finding the roots of a polynomial.

Given a polynomial, it only takes five steps to find every single root.

See if you can do some obvious factoring

Factor by Grouping

If it's a quadratic, normally factor it.

Basically, the sign changes show which ones are positive

If there are k sign changes, there are k - 2n positive roots (n is an integer, and k >= 2n)

To find the negative, plug in (-x) and find the sign changes, same thing applies with above

Don't forget about complex roots

Rational Root Theorem

The roots must be a fraction such that the factors of the final term are in the numerator and the factors of the first coefficient are in the denominator.

Find Bounds

If you use synthetic division (very useful, watch the KhanAcademy video), you will see the resulting polynomial.

If every term is positive, you know that term is the upper bound, because as you increase the x, everything is still positive.

If x is negative, and the terms alternate, you know it is a bound, because plugging in (-x) would make it all positive or negative.

Keep Narrowing the Bounds, and Testing Roots

As you continue narrowing the roots, finding the roots will be easier, as the possible range will get smaller and smaller

What do you do when you find a root? Well first, when you synthetically divide, you find the resulting polynomial. Then repeat the same steps and find the roots of that.

Hints:

- When plugging in a value that is negative, and another is positive, you know there is a root between them. (Also would work vice-versa, because a polynomial is continuous)
- Law of Ones (Most polynomials have a root at 1 or -1, because teachers want to be nice), With this method, that isn't as important, but it's a good place to start.

With these few steps and hints, finding roots will be a piece of cake.