Expressions, involving arithmetical numbers, variables and symbols of operations are called algebraic expressions. An algebraic expression is a combination of numbers, variables and arithmetical operations.
Constant: Quantity which has a fixed numerical value. For example: 0, 1, 2
Variable: Quantity which can take different numerical values. A variable is represented by a letter of the English alphabet such as a, b, c, x, y, z.
Algebraic expressions: A combination of constants and variables, connected by any or all of the four fundamental operations (+, -, ×, ÷).
Term: Each part of the expression along with its sign.
Monomial: An algebraic expression containing one term. For example, 6a2, 3x2y2.
Binomial: An algebraic expression containing two terms. For example, a2 + b2, 7xy + y2.
Trinomial: An algebraic expression containing three terms. For example, x2 + y2 + z2, x2 + 2xy + y2.
Polynomial: An algebraic expression in which variables do not occur in the denominator, exponents of variables are whole numbers and numerical coefficients of various terms are real numbers. For example, x3 - 2y2+ y - 7.
Factor: When two or more numbers or variables are multiplied, then each one of them and their product is called a factor of the product. A constant factor is a numerical factor while a variable is known as a literal factor.
Coefficient: In a term any one of the factors with the sign of the term is the coefficient of the product of the other factors. For example, in -3xy, coefficient of x is -3y.
Constant Term: Term which has no literal factor. For example, in 2x + 9y + 7 the constant term is 7.
Like and Unlike Terms: Terms having same literal factors are called like or similar terms and terms having different literal factors are called unlike terms.
Degree of a polynomial: Sum of the exponents of the variables in a term is called degree of the term. Degree of a polynomial is the same as the degree of its term or terms having the highest degree and non-zero coefficient.
Quadratic polynomial: A polynomial of degree 2. For example, x2 - 3x + 2.
Zero degree polynomial: Degree of a nonzero constant polynomial is taken as zero.
Zero polynomial: When all the coefficients of variables in the terms of a polynomial are zeros, the polynomial is called a zero polynomial and the degree of zero polynomial is not defined.
Zeros of a polynomial: Values of the variable for which the value of a polynomial in one variable is zero.
Addition and subtraction of polynomials: The sum of two (or more) like terms is a like term whose numerical coefficient is the sum of the numerical coefficients of the like terms.
The difference of two like terms is a like term whose numerical coefficient is the difference of the numerical coefficients of the like terms.
To add polynomials, add their like terms together. For example,
2x + 3x = 5x, 3x2y + 8x2y = 11x2y
To subtract a polynomial from another polynomial, subtract a term from a like term. For example,
9x2y2 - 5x2y2 = 42x2y
Multiplication of the polynomials: To multiply a monomial by a monomial, use laws of exponents and the rules of the signs. For example, 3a × a2b2 = 3a3b2
To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial.
To multiply a polynomial by another polynomial multiply each term of the one polynomial by each term of the other polynomial and simplify the the result by combining like terms.
Division of polynomials: To divide a monomial by another monomial, find the quotient of numerical coefficients and variables separately using laws of exponents and then multiply these quotients.
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
Process of division of a polynomial by another polynomial is done on similar lines as in Arithmetic after arranging the terms of both polynomials In decreasing powers of the variable common to both of them. If remainder is zero the divisor is a factor of dividend.
Dividend = Divisor x quotient + Remainder.
Factorization of polynomials: Factorization of polynomials is a process of writing the polynomial as a product of two (or more) polynomials. Each polynomial in the product is called a factor of the given polynomial.
HCF of polynomials: HCF of two or more given polynomials is the product of the polynomials of highest degree and greatest numerical coefficient each of which is a factor of each of the given polynomials.
LCM of polynomials: LCM of two or more polynomials is the product of the polynomials of the lowest degree and the smallest numerical coefficient which are multiples of the corresponding elements of each of the given polynomials.
Rational Expression: An algebraic expression which can be expressed in the form p/q where p is any polynomial and q is non-zero polynomial. A rational expression need not to be a polynomial. Every polynomial is a rational expression also.