Polynomial Equations and Functions
A polynomial equation or function with degree n has n number of solutions; for example, a polynomial with a degree of three has three solutions.
ex.
(x - 4)3(x + 7)2(x + 2) = 0
This equation has a degree of six
This can also be written as:
(x - 4)(x - 4)(x - 4)(x + 7)(x + 7)(x + 2) = 0
which implies that there are six solutions to the equation:
| x-4 = 0 | or | x-4 = 0 | or | x-4 = 0 |
| x = 4 | or | x = 4 | or | x = 4 or |
| x+7 = 0 | or | x+7 = 0 | ||
| x = -7 | or | x = -7 | or | |
| x+2 = 0 | ||||
| x = -2 |
The general solution for the equation is{4,-7,-2}, but it is also said that the equation has a solution 4 with multiplicity of three, a solution -7 with multiplicity of two.
Multiplicity is a repetitive solution to an equation, from the above example, the solution 4 has a multiplicity of three, meaning that the solution 4 is repeated three times.
Any solution of multiplicity p is counted p times.
Solving Polynomial Equations( with degree 3 or greater):