Class 7 maths In the previous class, we have learnt about algebraic expressions and their addition and subtraction. In this chapter we shall study multiplication and division of algebraic expressions in the form of monomials and binomials etc. CONSTANTS: A symbol having a fixed numerical value is called a constant. 3 For example: 9, - 3, , 3, p etc. are all constants. 8 Variables or Literals: A symbol which takes on various numerical values is known as a variable or a literal. We know that the perimeter of a square of side a is given by the formula, P = 4a. Here 4 is a constant, while a and P are variables. We may give any value to a and get the corresponding value of P. Area of circle given by đ??´ = đ?œ‹đ?‘&#x; % , where đ?œ‹ is constant and A, r are variables. Algebraic Expressions : A combination of constants and variables, connected by +, - , ´ and á is known as an algebraic expression. Examples :
(i) 6x + 8y is an expression containing 2 terms, namely 6x and 8y. (ii) 9x + 2y – 4xy is an expression containing 3 terms, namely, 9x, 2y and - 4xy. (iii) 4x5/2 – 7x2 + 6x2yz – 8z3 is an expression containing 4 terms, namely 4x5/2, - 7x2, 6x2yz and – 8z3.
Types of algebraic expressions: 1.
Monomial : An algebraic expression containing only one term, is called a monomial. Examples : 7x, 6y2, 3xy2z, - 5 are all monomials.
2.
Binomial : An algebraic expression containing 2 terms is called a binomial. Example: (i) 5x + 9y is a binomial having 2 terms, namely 5x and 9y. (ii) 2xy – 3 is a binomial having 2 terms, namely, 2xy and – 3. (iii) a2b – 3b2c is a binomial having 2 terms, namely, a2b and -3b2c.
3.
Trinomial: An algebraic expression containing 3 terms is called a trinomial. Example: (i) x + 2y – 3z is a trinomial having 3 terms, namely x, 2y and – 3. (ii) z2 – 2xy + 5 is a trinomial having 3 terms, namely, z2, - 2xy and 5.
4.
Multinomial: An algebraic expression containing more than 3 terms, is called a multinomial. Example : 3x2y + 2x3y – xyz + 2 is a multinomial, having 4 terms, namely, 3x2y, 2x3y, – xyz and 2.
Factors of A Term: When numbers and literals are multiple to form a product, then each quantity multiplied is called a factor of the product. A constant factor is called a numerical factor while a variable factor is called a literal factor. Thus, in – 7x2y, the numerical factor is – 7, and the literal factors are x, x2, y, xy and x2y. Constant Term: A term of the expression having no literal factor is called the constant term. Thus, in the expression x2 + y2 + xy – 3, the constant term is – 3. 25 Clearly, the expression, 3x4 – 2x3 + has no constant term. x Coefficients: Any factor of a term is called the coefficient of the product of other factors.
Example: (i) In 7xy, the numerical coefficient is 7 and literal coefficient is xy. Also, the coefficient of x is 7y and the coefficient of y is 7x. (ii) In – 9y2z, the numerical coefficient is – 9 and literal coefficient is y2z. Also, the coefficient of y2 is – 9z; the coefficient of z is – 9y2; the coefficient of 9y is – yz, and so on. Like Terms: Terms having same literal coefficients are called like terms, otherwise they are called unlike terms. 2 2 Example: (i) 7a2, – 3a2, a are like terms. 3 (ii) 6a2, – 8b2, 4ab are unlike terms. (iii) 8x, 8y2, 8x2 are unlike terms. POLYNOMIALS: An algebraic expression in which the variables involved have only non negative integral powers, is called a polynomial. Degree of a Polynomial in One Variable: The highest power of the variable in a polynomial of one variable is called the degree of the polynomial. Example: (i) 5x3 – 3x2 + 4x – 8 is a polynomial of degree 3. (ii) 6 + 5y2 – 7y4 is a polynomial of degree 4. (iii) 3z – 5z3 + 8z + 1 is a polynomial of degree 5. (iv) 3x +
3 is an expression but not a polynomial, since it contains a term x
3 in which power of x is – 1, which is not a non-negative integer. x (v) 2 + 5x3/2 + 7x2 is an expression but not a polynomial, since it contains a 3 term in which power of x is , which is not a non-negative integer. 2 (vi) z2 + 3 z + 9is an expression but not a polynomial, as it contains a term in 1 which power of z is , which is not a non-negative integer. 2
namely
Linear Polynomial: A polynomial of degree 1 is a called a linear polynomial. 3 Example: (a) 7 + 4x (b) 1 – 2z (c) + 7y are all linear polynomials. 2 Quadratic Polynomial: A polynomial of degree 2 is called a quadratic polynomial. Example: (a) 6z2 – 5z + 4 and (b) 3y2 + 7y – 2 are both quadratic polynomials. Cubic Polynomial: A polynomial of degree 3 is called a cubic polynomial. Example: (a) 3x3 – 2x2 + 1 (b) z3 – 3z2 + 5z + 7 are both cubic polynomials. Constant Polynomial: A polynomial having one term consisting of a constant only is a constant polynomial. The degree of a constant polynomial is 0. Thus, 7 is a constant polynomial.
Degree of a Polynomial in Two or More Variables: If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.