The roots (sometimes called zeroes or solutions) of a polynomial are the values of for which is equal to zero. Finding the roots of a polynomial is sometimes called solving the polynomial.
For example, if , then the roots of the polynomial are and , since both and are equal to zero.
Finding the root of a linear polynomial (a polynomial with degree one) is very straightforward. The formula for the root is (although calling this a formula is going a bit overboard).
The roots for a quadratic polynomial (a polynomial with degree two) is given by the formula
The formula for the roots of a cubic polynomial (a polynomial with degree three) is a bit more complicated while the formula for the roots of a quartic polynomial (a polynomial with degree four) would fill two blackboards!
What about fifth-degree (quintic) polynomials? What about polynomials with higher degrees?
About years ago, a young mathematician by the name of Henrik Abel proved that it is impossible to find a formula for the solutions of a quintic polynomial by adding, subtracting, multiplying, dividing and taking roots. More formally speaking, a quintic polynomial is not solvable by radicals.
The same is true for polynomials with higher degrees.