Mathwizurd.com is created by David Witten, a mathematics and computer science student at Vanderbilt University. For more information, see the "About" page.

Dec 30 Finding the roots of a polynomial

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.

1. See if you can do some obvious factoring

• Factor by Grouping

• If it's a quadratic, normally factor it.

2. Descartes' Law of Signs

• 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

3. 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.

4. 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.

5. 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.

David Witten