What is Polynomial | Various kinds of Polynomials
Constant
A symbol
having a fixed numerical value is called a constant
Examples: 3, -6, 5/6, 0.23, π, etc. are all constants.
Variable
A symbol which may be assigned different
numerical values us known as a variable.
Example: We know that the area of a circle is given by the
formula A = πr2, where r is the radius of the circle.
Here, π is constant, while A and r are variables.
Algebraic Expression
A combination of constants and
variables, connected by some or all of the operations +, −, ×, and ÷ is known
an algebraic expression.
Examples: x +2, x2+y2, x – 1/x, etc.
Terms of an Algebraic Expression
The several parts of an
algebraic expression separated by + or – operations are called the terms of the
expression.
Examples:
(i). 3 + 4x – 7xy2 + 9xy is an
algebraic expression containing four terms, namely, 3, 4x, 7xy2 and
9xy
(ii). X3 + 3x2y – 5xy2 + y3
– 6 is an algebraic expression containing five terms, namely, X3, 3x2y,
5xy2, y3 and 6.
Polynomial
An algebraic expression in which the variables
involved have only non-negative integral powers is called a polynomial.
Examples:
(i). x3 + 5x2 - 2x + 3 is a
polynomial in one variable x.
(ii). 2p4 + 5p3 – 2p2
+ p – 5 is a polynomial in one variable y.
(iii).
3 + x – x3/2 + 2x3 is an expression but not a polynomial
since it contains a term containing x3/2, where 3/2 is not a
non-negative integer.
Degree of Polynomial in one variable
In case of a
polynomial in one variable, the highest power of the variable is called the
degree of the polynomial.
Examples:
(i). 3x – 4 is a polynomial in x of degree 1.
(ii). 2m2 + 6m – 5 is a polynomial in m of degree 2.
Degree of a Polynomial in two or more variables
In case of
polynomials in more than one variable, the sum of the powers of the variables
in each term is taken up and the highest sum so obtained is called the degree
of the polynomial.
Examples:
(i). 2x3 + 3x3y – 4xy + 3x –
5 is a polynomial in x and y of degree 4.
(ii). 3m4n2 + 2m2n2 – 5m +
4n – 2 is a polynomial in m and n of degree 6
Various Type of Polynomials
1. Polynomials of Various Degrees
(i). linear polynomial
A polynomial of degree 1 is called a linear polynomial.
Examples: (a). 2x – 5 is a linear polynomial in x.
(b). 3p + 4 is a linear polynomial in p.
(c). x + 2y +5 is a linear polynomial in a x and y.
(ii). Quadratic Polynomial
A polynomial of degree 2 is called a quadratic polynomial.
Examples: (a). x2 + 5x – 2 is a quadratic polynomial in x.
(b). 2z2 – 3z + 2 is a quadratic polynomial in z.
(c). ab + bc + ca is a quadratic polynomial in a, b and c.
(iii). Cubic Polynomial
A polynomial of degree 3 is called a cubic polynomial.
Examples: (a). 2x3 + 3x2 – 5x + 2 is cubic polynomial in x.
(b). x2y – 2xy + 3 is cubic polynomial in x and y.
(iv). Biquadratic Polynomial
A polynomial of degree 4 is called a biquadratic polynomial.
Examples:
(a). x4 + 3x3 – 6x2 + 5x + 2 is biquadratic polynomial in x.
(b). x3y + xy2 + 2xy – 5 is biquadratic polynomial in x and y.
2. Number of terms in a Polynomial:
(i). Monomial
A polynomial containing one nonzero term is called a monomial.
Examples: 2, 3x, xy2, 5x3y etc.
(ii). Binomial
A polynomial containing two nonzero terms is called a binomial.
Examples: (5x + 3), (x2 + y2), (x – 3y) etc.
(iii). Trinomial
A polynomial containing three nonzero term is called a trinomial.
Examples: (x2 + 3x +2), (3y – 2xy + 3xy2) etc.
Constant polynomial
A polynomial containing one nonzero term, consisting of a constant is called a constant polynomial. The degree of nonzero of constant polynomial is zero.
Examples: 2, - 4, 2/3 etc.
Zero Polynomial
A polynomial containing one term, namely zero only, is called a zero polynomial. The degree of a zero polynomial is not defined.
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