CLASS 8 MATHS Algebraic Expression Introduction : Algebra like every other branch of Mathematics should be studied more as a subject for mental discipline than for anything else. An intelligent grasp of principles, therefore, is to be chiefly aimed at and not the more learning by rote of a certain number of rules of a certain number of rules eith some readiness in their application. Quantity: Anything that can be represented by a number is called a quantity Ex : Time, weight, money, distance etc. Quantity is also often used in the sense of number, integral or fractional. An algebraic expression also is some times called a quantity. What is Algebra? Algebra, like arithmetic is a science of numbers with the distinction that the numbers in algebra are generally denoted by letters instead of by figures. Symbols : The letters of the alphabet a, b, c, ..... are used to denote numbers and the signs +, –, ×, ÷, =, ..... and c are used either to denote operations to be performed upon the number to whcih they are attached or as abbreviations. These letters and signs are called symbols. Expression ; Term : Any intelligible collection of letters, numerals and signs of operations is called an Algebraical expression. Such collection is also sometimes called an Algebraical quantity, or briefly, quantity. The parts of an algebraical expression that are connected by the signs + are – are called its terms. Ex : 5a + ab + c × d – 8c × f × g is an algebraical expression of which terms are 5a, ab + c × d, 8c × f ÷ g is an algebraical expression of which terms of 5a, ab + c × d, 8c × f ÷ g. Expressions are either simple or compound. A simple expression is one which has no parts no parts connected by the sign + or –, i.e., which consists of only one term, as 3ab,
5x 4x × 3y and is , 6y 2z
also called a monomial. A compound expression consists of two or more terms, if it consist of two terms, as 2a + 5bcd, 5x +
6y , it is called a binomial. 7z
f of three terms, as a + bc + 8efg, x × y ÷ z + a × c –
8b , a Trinomial. 9e
And if of more than three terms, a Multinomial or a polynomial. Functions ; Variables : Any expression involving a letter is called a function of that letter. Thus, x3 + 5x + 8 is a function of x ; a2 + ab + b2 is a function of a and b ; a3 + b3 + c3 + 2abc is a function of a, b and c and so on. The letters of which a function consists are called its variables. Thus, x2 + 5xy + y2 is a function of which the variables are x and y.
Factor : If any number be equal to the product of two more numbers, each of the latter is called a factor of the former. Thus, 3, 5 and 7 are the factors of 105, ! 105 = 3 × 5 × 7 Similarly, 3, a, b and x are the factors of 3abx, because 3abx = 3 × a × b × x. Coefficient : The number expressed in figures or symbol which stands before an algebraical quantity as a multiplier, is call its coefficient. Thus, in 5abc, 5 is the coefficient of abc, 5a is the coefficient of bc and 5ab is the coefficient of c. Like and Unlike terms : Terms or simple expressions are said to be like when they do not differ at all or differ only in their numerical coefficients; otherwise they are called unlike. Thus, 3ax2y2 and 5ax2y5 are like terms, whereas 3ax2y5 and 5ax2y4 are unlike, similarly, abc, 5axbd, 7a2b8 and c2d8x are all unlike. Polynomials : An algebraic expression in which the powers of the variables involved are non-negative integers, is called a polynomial. Polynomials in one variable and their degree : An algebraic expression involving only one variable in which the powers of the variable are non-negative integers, is a polynomial in that variable. Degree : The highest power of the variable in a polynomial is called its degree. Examples : i) 3x + 7 is a polynomial in x of degree 1. ii) 2y2 – 5y + 1 is a polynomial in y of degree 2. iii) iv)
z3 + 4z2 + 3z – 6 is a polynomial in z of degree 3. 1 1 is not a polynomial, since = x -1, i.e., power of x is a negative integer. Note x x 1 that x + is a binomial expression but it not a polynomial. x x+
Polynomials in two or more variables : An algebraic expression involving two or more variables with non-negative integral powers is called a polynomial in these variabels. The degree of any term of a polynomial is the sum of the powers of all the variables in that term. Degree : The degree of the highest degree term in a polynomial is called the degree of the polynomial.