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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. 


  • 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

Fast Coprime Checker