Some of the important properties of polynomials along with some important polynomial theorems are as follows:
Property 1: Division Algorithm
If a polynomial P(x) is divided by a polynomial G(x) results in quotient Q(x) with remainder R(x), then,
P(x) = G(x) • Q(x) + R(x)
Where R(x)=0 or the degree of R(x) < the degree of G(x)
Property 2: Bezout’s Theorem
Polynomial P(x) is divisible by binomial (x – a) if and only if P(a) = 0.
Property 3: Remainder Theorem
If P(x) is divided by (x – a) with remainder r, then P(a) = r.
Property 4: Factor Theorem
A polynomial P(x) divided by Q(x) results in R(x) with zero remainders if and only if Q(x) is a factor of P(x).
Property 5: Intermediate Value Theorem
If P(x) is a polynomial, and P(x) ≠ P(y) for (x < y), then P(x) takes every value from P(x) to P(y) in the closed interval [x, y].
Property 6
The addition, subtraction and multiplication of polynomials P and Q result in a polynomial where,
Degree(P ± Q) ≤ Degree(P or Q)
Degree(P × Q) = Degree(P) + Degree(Q)
Property 7
If a polynomial P is divisible by a polynomial Q, then every zero of Q is also a zero of P.
Property 8
If a polynomial P is divisible by two co-prime polynomials Q and R, then it is divisible by (Q • R).
Property 9
If P(x) = a0 + a1x + a2x2 + …… + anxn is a polynomial such that deg(P) = n ≥ 0 then, P has at most “n” distinct roots.
Property 10: Descartes’ Rule of Sign
The number of positive real zeroes in a polynomial function P(x) is the same or less than by an even number as the number of changes in the sign of the coefficients. So, if there are “K” sign changes, the number of roots will be “k” or “(k – a)”, where “a” is some even number.
Property 11: Fundamental Theorem of Algebra
Every non-constant single-variable polynomial with complex coefficients has at least one complex zero.
Property 12
If P(x) is a polynomial with real coefficients and has one complex zero (x = a – bi), then x = a + bi will also be a zero of P(x). Also, x2 – 2ax + a2 + b2 will be a factor of P(x).
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