Algebraic Expressions and Identities

Transcription

1 9 Algebraic Epressions and Identities introduction In previous classes, you have studied the fundamental concepts of algebra, algebraic epressions and their addition and subtraction. In this chapter, we shall review these topics and shall learn multiplication and division of algebraic epressions. We shall also learn about algebraic identities and their applications. fundamental concepts In algebra, we use two types of symbols constants and variables (literals). Constant. A symbol which has a fi ed value is called a constant. For eample, each of 7,,, 7,,, π etc. is a constant. 5 Variable. A symbol which can be given various numerical values is called a variable or literal. For eample, the formula for circumference of a circle is C = πr, where C is the length of the circumference of the circle and r is its radius. Here, π are constants and C, r are variables (or literals). Algebraic epression A collection of constants and literals (variables) connected by one or more of the operations of addition, subtraction, multiplication and division is called an algebraic epression. The various parts of an algebraic epression separated by + or sign are called terms of the algebraic epression. Various types of algebraic epressions are : Monomial. An algebraic epression having only one term is called a monomial. Binomial. An algebraic epression having two terms is called a binomial. Trinomial. An algebraic epression having three terms is called a trinomial. Multinomial. An algebraic epression having two or more terms is called a multinomial. For eample: Algebraic epression No. of terms Name Terms (i) 7 y 1 Monomial 7 y (ii) 5 7 y Binomial 5 7 y, y y (iii) y + 5z 11 + Trinomial y, 5z 11, (iv) Multinomial 9 5,, 4,

2 Algebraic Epressions and Identities 15 Remark Multiplication and division do not separate the terms of an algebraic epression. Thus, 7 y is one term while 7 + y has two terms. Factors. Each of the quantity (constant or literal) multiplied together to form a product is called a factor of the product. A constant factor is called a numerical factor and any factor containing only literals is called a literal factor. In 7y, the numerical factor is 7 and the literal factors are, y, y, y and y. Constant term. The term of an algebraic epression having no literal factors is called its constant term. In the epression y 5 + 7, 7 is the constant term, while the epression has no constant term. Coefficient. Any factor of a (non-constant) term of an algebraic epression is called the coefficient of the remaining factor of the term. In particular, the constant part is called the numerical coefficient or simply the coefficient of the term and the remaining part is called the literal coefficient of the term. Consider the epression 7p q 5p q p +. In the term 5p q: the numerical coefficient = 5, the literal coefficient = p q, the coefficient of p = 5q, the coefficient of 5p = pq, the coefficient of q = 5p etc. Note. When we write, we mean 1. Thus, if no coefficient is written before a literal, then the coefficient is always taken as 1. Like and unlike terms. The terms having same literal coefficients are called like terms; otherwise, they are called unlike terms. For eample: (i) 5 yz, yz, yz are like terms 5 (ii) 7ab, a b, ab are unlike terms. Polynomial An algebraic epression is called a polynomial if the powers of the variables involved in it in each term are non-negative integers. A polynomial may contain any number of terms, one or more than one. Take the sum of the powers of the variables in each term; the greatest sum is the degree of the polynomial. For eample: (i) + 5 is a polynomial of two terms; degree 1 (ii) 8 y 7y + 9y + is a polynomial of four terms; degree (iii) is a polynomial of three terms; degree 8 (iv) + 5y + y + 5 y + 9 is a polynomial of five terms; degree (v) is not a polynomial because in term = 1, the power of is a negative integer.

3 154 Learning Mathematics VIII Addition and subtraction of algebraic epressions In class VII, you have learnt how to add and subtract algebraic epressions. Let us recall these operations. Addition of algebraic epressions To add two or more algebraic epressions, we may use horizontal method or column method. Horizontal Method: In horizontal method we collect different groups of like terms and then find the sum of like terms in each group. Column Method: In column method we write each epression to be added in a separate row. While doing so we write like terms one below the other and add them. Eample 1. Add the following: (i) 5y + 4y 1, 7y 9y + 5, 7y 4 + y 1 (ii) 7y + 5yz z, 4yz + 9z 4y, z + 5 y (i) Horizontal method: ( 5y + 4y 1) + (7y 9y + 5) + (7y 4 + y 1) = 4 5y 9y + 7y + 4y + 7y + y = 7y + 1y 9, which is the required sum (ii) Horizontal method: 5y + 4y 1 + 9y + 7y y + y 1 7y + 1y 9, which is the required sum. (7y + 5yz z) + (4yz + 9z 4y) + ( z + 5 y) = 7y y + 5yz + 4yz z + 9z z + 5 4y (Note z is same as z) = 5y + 9yz + z + 5 4y, which is the required sum. 7y + 5yz z + 4yz + 9z 4y + y z + 5 (Note z is same as z) 5y + 9yz + z + 5 4y, which is the required sum. Subtraction of algebraic epressions To subtract algebraic epressions change the sign of each of the algebraic epression to be subtracted and then add. We may use horizontal method or column method. Eample. Subtract: (i) 5 8 from (ii) 5 4y + 6y from 7 4y + 8y + 5 y. Arrange like terms in such a way that they are one below the other

4 Algebraic Epressions and Identities 155 (i) Horizontal method: ( ) (5 8) = = = (ii) Horizontal method: (7 4y + 8y + 5 y) (5 4y + 6y ) = 7 4y + 8y + 5 y 5 + 4y 6y + = 7 5 4y + 8y + 4y + 5 y 6y + = 4y + 1y + 5 9y + 7 4y + 8y + 5 y 5 4y + 6y + + 4y + 1y + 5 9y + Change the sign of each term to be subtracted and then add Eercise Identify the terms, their numerical as well as literal coefficients in each of the following epressions: (i) 5yz zy (ii) (iii) 4 y 4 y z + z (iv) pq + qr rp (v) + y y (vi) 0.a 0.6ab + 0.5b. Identify monomials, binomials and trinomials from the following algebraic epressions: (i) 5 p q r (ii) y z (iii) + 7 5a b + c (iv) (v) 7 5 (vi) 5p q p q y (vii) m n + 5m (viii) 9a b c 5a + 1 (i) Identify which of the following epressions are polynomials. If so write their degrees: (i) (ii) (iii) 4a b ab 4 + 5ab + (iv) y y + 5y + 4. Add the following epressions: (i) ab bc, bc ca, ca ab (ii) a b + ab, b c + bc, c a + ac (iii) p q pq + 4, 5 + 7pq p q (iv) l + m, m + n, n + l, lm + mn + nl (v) , 5 + 4, , 6 1

5 156 Learning Mathematics VIII 5. Subtract: (i) 4a 7ab + b + 1 from 1a 9ab + 5b (ii) y + 5yz 7z from 5y yz z + 10yz (iii) 4p q pq + 5pq 8p + 7q 10 from 18 p 11q + 5pq pq + 5p q (iv) 9y + y from 7 10y y Subtract + 5y 4z + from the sum of 5 y + 7z and 7y 6 + 5z Subtract the sum of + 5y + 7y + and 4y y + 7 from 9 8y + 11y. 8. What must be subtracted from a 5ab b to get 5a 7ab b + a? 9. The two adjacent sides of a rectangle are y and + y. Find its perimeter. 10. The perimeter of a triangle is 7p 5p + 11 and two of its sides are p + p 1 and p 6p +. Find the third side of the triangle. Multiplication of algebraic epressions To multiply two or more algebraic epressions, we should follow two rules: (i) The product of two factors with like signs is positive, and the product of two factors with unlike signs is negative. i.e. (+) (+) = (+), ( ) ( ) = (+), (+) ( ) = ( ), ( ) (+) = ( ). (ii) If is a literal (variable) and m, n are positive integers, then m n = m + n. Multiplication of two or more monomials Product of two or more monomials = (product of their numerical coefficients) (product of their literal coefficients) Eample 1. Find the product of: (i) y and y (ii) 7 ab, 4a b c and bc (i) ( y ) ( y ) = ( ( )) ( y y ) = 6 ( ) (y y ) = 6 y 5 (ii) (7ab) ( 4a b c ) ( bc ) = (7 ( 4) ( )) (a a ) (b b b) (c c ) = 84a b 4 c 5. Eample. Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively: (i) (, 4y) (ii) (9y, 4y ) (iii) (4ab, 5bc) (iv) (l m, lm ) (i) Area of rectangle = length breadth = () (4y) = ( 4) y = 1y. (ii) Area of rectangle = (9y) (4y ) = (9 4) y y = 6y (iii) Area of rectangle = (4ab) (5bc) = (4 5) ab bc = 0ab c