Hi friends, please explain polynomials' basics and some important points in simple way. Please help me. And also explain roots to rational numbers in an easy way.
Answers
Answer:
What is a Polynomial?
Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable). Based on the numbers of terms present in the expression, it is classified as monomial, binomial, and trinomial. Examples of constants, variables and exponents are as follows:
Constants. Example: 1, 2, 3, etc.
Variables. Example: g, h, x, y, etc.
Exponents: Example: 5 in x5 etc.
Notation
The polynomial function is denoted by P(x) where x represents the variable. For example,
P(x) = x2-5x+11
If the variable is denoted by a, then the function will be P(a)
Degree of a Polynomial
The degree of a polynomial is defined as the highest degree of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial.
Terms of a Polynomial
The terms of polynomials are the parts of the equation which are generally separated by “+” or “-” signs. So, each part of a polynomial in an equation is a term. For example, in a polynomial, say, 2x2 + 5 +4, the number of terms will be 3. The classification of a polynomial is done based on the number of terms in it.
Polynomial Terms Degree
P(x) = x3-2x2+3x+4 x3, -2x2, 3x and 4 3
Types of Polynomials
Polynomials are of 3 different types and are classified based on the number of terms in it. The three types of polynomials are:
Monomial
Binomial
Trinomial
These polynomials can be combined using addition, subtraction, multiplication, and division but is never division by a variable. A few examples of Non Polynomials are: 1/x+2, x-3
Monomial
A monomial is an expression which contains only one term. For an expression to be a monomial, the single term should be a non-zero term. A few examples of monomials are:
5x
3
6a4
-3xy
Binomial
A binomial is a polynomial expression which contains exactly two terms. A binomial can be considered as a sum or difference between two or more monomials. A few examples of binomials are:
– 5x+3,
6a4 + 17x
xy2+xy
Trinomial
A trinomial is an expression which is composed of exactly three terms. A few examples of trinomial expressions are:
– 8a4+2x+7
4x2 + 9x + 7
Monomial Binomial Trinomial
One Term Two terms Three terms
Example: x, 3y, 29, x/2 Example: x2+x, x3-2x, y+2 Example: x2+2x+20
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) = a0 + a1x + a2x2 + …… + anxn is a polynomial such that deg(P) = n ≥ 0 then, P has at most “n” distinct roots.