A polynomial is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient. The simplest polynomials have one variable. A one-variable (univariate) polynomial of degree n has the following form:
where the a's represent the coefficients and x represents the variable.
for all complex numbers x, the above expression can be simplified to:
When an nth-degree univariate polynomial is equal to zero, the result is a univariate polynomial equation of degree n:
There may be several different values of x, called roots, that satisfy a univariate polynomial equation. In general, the higher the order of the equation (that is, the larger the value of n), the more roots there are.
Noted that power of x must be positive and otherwise it’s not polynomial
for example : a x^3 +b x^-2 +c x = 0 is not polynomial
Step-by-step explanation:
Properties
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)
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).
Learn More: Factor Theorem
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].
Learn More: Intermediate Value Theorem
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 coprime polynomials Q and R, then it is divisible by (Q • R).
Property 9
If P(x) xn is a polynomial such that deg(P) = n ≥ 0 then, P has at most “n” distinct roots.
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 root.
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).
Polynomial Equations
The polynomial equations are those expressions which are made up of multiple constants and variables. The standard form of writing a polynomial equation is to put the highest degree first then, at last, the constant term. An example of a polynomial equation is:
b = a4 +3a3 -2a2 +a +1
Polynomial Functions
Property 10: Descartes’ Rule of Sign
A polynomial function is an expression constructed with one or more terms of variables with constant exponents. If there are real numbers denoted by a, then function with one variable and of degree
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