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A symbol that can be assigned different numerical values is known as a variable. Variables are generally denoted by x, y, z, p, q, r, s, etc.
A symbol having a fixed value is called a constant, e.g. 8, 5, 9, p, a, b, c, etc. are constants.
Note: The values of constants remain the same throughout a particular situation, but the value of a variable can keep changing.
A combination of constants and variables, connected by some or all the basic operations +, –, x, is called an algebraic expression.
Example: is an algebraic expression.
Various parts of an algebraic expression separated by (+) or (–) operations are called terms.
Examples: (i) In the above algebraic expression, terms are 7, 8x, –6x2y and
(ii) Various terms of 3p4 – 6q2 + 8r3s – 2pq + 6s3 are: 3p4, –6q2, 8r3s, –2pq and 6s3
An algebraic expression in which the variables involved have non-negative integral powers is called a polynomial.
Examples: (i) x3 + x2 – 4x – 7;
(ii) 3p3 + 5p – 9;
(iii) x2 + 2x; etc. are all polynomials
Note: I. Each variable in a polynomial has a whole number as its exponent.
II. Each term of a polynomial has a coefficient.
I. In the case of a polynomial involving one variable, the highest power of the variable is called the degree of the polynomial.
Example: The degree of x5 – 2x3 + x is 5.
II. In the case of a polynomial involving more than one variable, the highest sum of exponents of variables in any term is called the degree of the polynomial.
Example: The degree of p2 – 6p6q + 5p2q3 – 3q4 is 7
(∵ The term –p6q has the sum of exponents of p and q as 6 + 1, i.e. 7)
A [On the basis of the number of terms]
(i) Monomial: A polynomial containing only one non-zero term is called a monomial.
(ii) Binomial: A polynomial containing two non-zero terms is called a binomial.
(iii) Trinomial: A polynomial containing three non-zero terms is called a trinomial.
Note: I. A monomial containing a constant only is called a constant polynomial.
II. A monomial containing its term as zero only is called a zero polynomial.
III. If we add polynomials, we get a polynomial.
IV. If we multiply polynomials, we get a polynomial.
B [On the basis of its degree]
(i) Linear Polynomial: A polynomial of degree 1 is called a linear polynomial.
(ii) Quadratic Polynomial: A polynomial of degree 2 is called a quadratic polynomial.
(iii) Cubic Polynomial: A polynomial of degree 3 is called a cubic polynomial.
(iv ) Biquadratic Polynomial: A polynomial of degree 4 is called a biquadratic polynomial.