What is the necessary condition for division of two polynomials in one variable? please say the definition :-)
Answers
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.
PolynomialDegreeExampleConstant or Zero Polynomial06Linear Polynomial13x+1Quadratic Polynomial24x2+1x+1Cubic Polynomial36x3+4x3+3x+1Quartic Polynomial46x4+3x3+3x2+2x+1
Example: Find the degree of the polynomial 6s4+ 3x2+ 5x +19
Solution:
The degree of the polynomial is 4.
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.
PolynomialTermsDegreeP(x) = x3-2x2+3x+4x3, -2x2, 3x and 43
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:
- Monomia
- lBinomial
- 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:
5x36a4-3xy
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 + 17xxy2+xy
Trinomial
A trinomial is an expression which is composed of exactly three terms. A few examples of trinomial expressions are:
– 8a4+2x+74x2 + 9x + 7
MonomialBinomialTrinomialOne TermTwo termsThree termsExample: x, 3y, 29, x/2Example: x2+x, x3-2x, y+2Example: x2+2x+20
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