General Form of a Polynomial
A univariate polynomial is a mathematical expression involving a sum of powers in one variable multiplied by coefficients. Equation A shows the general form of an nth-order polynomial.
| P(x) = a0 + a1x + a2x2 + … + anxn | (A) |
where P(x) is the nth-order polynomial, the highest power n is the order of the polynomial if an ≠ 0, a0, a1, …, an are the constant coefficients of the polynomial and can be either real or complex.
You can rewrite Equation A in its factored form, as shown in Equation B.
| P(x) = an(x – r1)(x – r2) … (x – rn) | (B) |
where r1, r2, …, rn are the roots of the polynomial.
The root ri of P(x) satisfies the following equation.
 |
(C) |
In general, P(x) might have repeated roots, such that Equation D is true.
 |
(D) |
The following conditions are true for Equation D:
- r1, r2, …, rl are the repeated roots of the polynomial
- ki is the multiplicity of the root ri, i = 1, 2, …, l
- rl + 1, rl + 2, …, rl + j are the non-repeated roots of the polynomial
- k1 + k2 + … + kl + j = n
A polynomial of order n must have n roots. If the polynomial coefficients are all real, the roots of the polynomial are either real or complex conjugate numbers.
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