Download MultiplicAtion of AlgebrAic expressions

9
Algebraic Expressions
and Identities
introduction
In previous classes, you have studied the fundamental concepts of algebra, algebraic expressions and
their addition and subtraction. In this chapter, we shall review these topics and shall learn multiplication
and division of algebraic expressions. 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 fixed value is called a constant.
For example, each of 7, – 3, 2 , – 7 ,
5
2
2 , 2 – 3 , π etc. is a constant.
Variable. A symbol which can be given various numerical values is called a variable or literal.
For example, the formula for circumference of a circle is C = 2πr, where C is the length of the
circumference of the circle and r is its radius. Here 2, π are constants and C, r are variables
(or literals).
Algebraic expression
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 expression.
The various parts of an algebraic expression separated by ‘+ or –’ sign are called terms of the algebraic
expression. Various types of algebraic expressions are :
Monomial. An algebraic expression having only one term is called a monomial.
Binomial. An algebraic expression having two terms is called a binomial.
Trinomial. An algebraic expression having three terms is called a trinomial.
Multinomial. An algebraic expression having two or more terms is called a multinomial.
For example:
Algebraic expression
2 3
(i) – 7x y
(ii) 5x2y – 7x
No. of terms
Name
Terms
1
Monomial
– 7x2y3
2
Binomial
5x2y, – 7x
y
y
(iii) – 3xy3 + 5xz2 + 11
3
(iv) 9x5 – 3x2 + 4 – 33
4
Trinomial
2
x
– 3xy3, 5xz2, 11
2
Multinomial
9x5, – 3x2, 4, – 33
x
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153
Remark
Multiplication and division do not separate the terms of an algebraic expression. Thus,
– 7x2y3 is one term while –7x2 + y3 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 – 7xy2, the numerical factor is – 7 and the literal factors are x, y, y2, xy and xy2.
Constant term. The term of an algebraic expression having no literal factors is called its constant
term.
In the expression – 3x2y3 + 5 – 7, – 7 is the constant term,
x
while the expression 9x5 – 3x2 + 113 has no constant term.
x
Coefficient. Any factor of a (non-constant) term of an algebraic expression 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 expression 7p3q2 – 5p2q – 3p + 2. In the term – 5p2q:
the numerical coefficient = – 5,
the literal coefficient = p2q,
the coefficient of p2 = – 5q,
the coefficient of 5p = – pq,
the coefficient of – q = 5p2 etc.
Note. W
hen we write x, we mean 1x. 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 example:
(i)5x2yz, – 3x2yz, 3 yzx2 are like terms
5
(ii)7ab, – 3a2b, 2 ab2 are unlike terms.
3
Polynomial
An algebraic expression 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 example:
(i) 3x + 5 is a polynomial of two terms; degree 1
(ii) 8x2y – 7xy2 + 9xy + 3 is a polynomial of four terms; degree 3
(iii) 5x2 + 4x + 7 is a polynomial of three terms; degree 2
(iv) 3x + 5y + 8 xy + 5x2y + 9 is a polynomial of five terms; degree 3
3
(v) 9x +
3
x
+ 4 is not a polynomial because in term 3 = 3x–1, the power of x is a negative integer.
x
Learning Mathematics–VIII
154
Addition and subtraction of
algebraic expressions
In class VII, you have learnt how to add and subtract algebraic expressions. Let us recall these
operations.
Addition of algebraic expressions
To add two or more algebraic expressions, 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 expression to be added in a separate row. While
doing so we write like terms one below the other and add them.
Example 1. Add the following:
(i)3x2 – 5xy + 4y2 – 1, 7y2 – 9xy + 5, 7xy – 4x2 + y2 – 13
(ii)7xy + 5yz – 3zx, 4yz + 9zx – 4y, – 3xz + 5x – 2xy
Solution.
(i)Horizontal method:
(3x2 – 5xy + 4y2 – 1) + (7y2 – 9xy + 5) + (7xy – 4x2 + y2 – 13)
= 3x2 – 4x2 – 5xy – 9xy + 7xy + 4y2 + 7y2 + y2 – 1 + 5 – 13
= – x2 – 7xy + 12y2 – 9, which is the required sum
Column method:
Arrange like terms in such a way
3x2 – 5xy + 4y2 – 1
that they are one below the
other
+ – 9xy + 7y2 + 5
2
2
+ – 4x + 7xy + y – 13
– x2 – 7xy + 12y2 – 9, which is the required sum.
(ii)Horizontal method:
(7xy + 5yz – 3zx) + (4yz + 9zx – 4y) + (– 3xz + 5x – 2xy)
= 7xy – 2xy + 5yz + 4yz – 3zx + 9zx – 3zx + 5x – 4y
= 5xy + 9yz + 3zx + 5x – 4y, which is the required sum.
Column method:
7xy + 5yz
+ 4yz
+ – 2xy
5xy + 9yz
–
+
–
+
(Note xz is same as zx)
3zx
9zx – 4y
3zx + 5x
(Note zx is same as xz)
3zx + 5x – 4y, which is the required sum.
Subtraction of algebraic expressions
To subtract algebraic expressions change the sign of each of the algebraic expression to be subtracted
and then add. We may use horizontal method or column method.
Example 2. Subtract:
(i)5x3 – 3x2 – 8 from 2x3 – 5x2 – 11x + 2
(ii)5x2 – 4y2 + 6y – 3 from 7x2 – 4xy + 8y2 + 5x – 3y.
Algebraic Expressions and Identities
155
Solution.
(i)Horizontal method:
(2x3 – 5x2 – 11x + 2) – (5x3 – 3x2 – 8)
= 2x3 – 5x2 – 11x + 2 – 5x3 + 3x2 + 8
= 2x3 – 5x3 – 5x2 + 3x2 – 11x + 2 + 8
= – 3x3 – 2x2 – 11x + 10.
Column method:
2x3 – 5x2 – 11x + 2
– 8
5x3 – 3x2
–
+
+
3
2
– 3x – 2x – 11x + 10
Change the sign of each term
to be subtracted and then add
(ii)Horizontal method:
(7x2 – 4xy + 8y2 + 5x – 3y) – (5x2 – 4y2 + 6y – 3)
= 7x2 – 4xy + 8y2 + 5x – 3y – 5x2 + 4y2 – 6y + 3
= 7x2 – 5x2 – 4xy + 8y2 + 4y2 + 5x – 3y – 6y + 3
= 2x2 – 4xy + 12y2 + 5x – 9y + 3
Column method:
7x2 – 4xy + 8y2 + 5x – 3y
– 4y2 + 6y – 3
5x2
– + – +
2
2x – 4xy + 12y2 + 5x – 9y +3
Exercise 9.1
1. Identify the terms, their numerical as well as literal coefficients in each of the following
expressions:
(ii) 1 + x + x2
(i) 5xyz2 – 3zy (iii) 4x2y2 – 4x2y2z2 + z2
y
(v) x + – xy
(iv) 3 – pq + qr – rp
2
(vi) 0.3a – 0.6ab + 0.5b
2
2. Identify monomials, binomials and trinomials from the following algebraic expressions:
(i) 5 p × q × r2 (ii) 3x2 × y ÷ 2z (iii) – 3 + 7x2
2
2
(iv) 5a − 3b + c (v) 7x5 – 3x 2
(vi) 5p ÷ 3q – 3p2 × q2
y
(vii) m3 – 2n2 + 5m – 2 2
(ix) 5x4 + 2x + 3x + 1
(viii) – 9a3b3c3 – 5a2 + 1
3
5
3. Identify which of the following expressions are polynomials. If so write their degrees:
(i) 2 x4 –
5
(ii)7x3 – 32 +
3 x2 + 5x – 1
5
x
(iii)4a3b2 – 3ab4 + 5ab + 2 3
(iv)2x2y – 3 + 5y3 +
xy
3
4. Add the following expressions:
(i)ab – bc, bc – ca, ca – ab
(ii) a – b + ab, b – c + bc, c – a + ac
(iv) l2 + m2, m2 + n2, n2 + l2, 2lm + 2mn + 2nl
(iii)2p2q2 – 3pq + 4, 5 + 7pq – 3p2q2
(v)4x3 – 7x2 + 9, 3x2 – 5x + 4, 7x3 – 11x + 1, 6x2 – 13x
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Learning Mathematics–VIII
5. Subtract:
(i) 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3
(ii) 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz
(iii) 4p2q – 3pq + 5pq2 – 8p + 7q – 10 from 18 – 3p – 11q + 5pq – 2pq2 + 5p2q
(iv) 3x2 – 9xy + y2 from 7x2 – 10xy – y2 + 5
6. Subtract 3x + 5y – 4z + 3 from the sum of 5x – 3y + 7z – 2 and 7y – 6x + 5z + 11.
7. Subtract the sum of 3x2 + 5xy + 7y2 + 3 and 2x2 – 4xy – 3y2 + 7 from 9x2 – 8xy + 11y2.
8. What must be subtracted from 3a2 – 5ab – 2b2 – 3 to get 5a2 – 7ab – 3b2 + 3a?
9. The two adjacent sides of a rectangle are 3x2 – 2y2 and x2 + 3xy. Find its perimeter.
10. The perimeter of a triangle is 7p2 – 5p + 11 and two of its sides are p2 + 2p – 1 and 3p2 – 6p + 3.
Find the third side of the triangle.
Multiplication of algebraic expressions
To multiply two or more algebraic expressions, 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 x is a literal (variable) and m, n are positive integers, then
x m × x n = x m + n.
Multiplication of two or more monomials
Product of two or more monomials = (product of their numerical coefficients) × (product of their
literal coefficients)
Example 1. Find the product of:
(i)2x2y3 and – 3xy2 (ii)7ab, – 4a2b2c2 and – 3bc3
Solution.
(i)(2x2y3) × (– 3xy2)= (2 × (– 3)) × (x2y3 × xy2)
= – 6 × (x2 × x) × (y3 × y2)
= – 6x3y5
(ii)(7ab) × (– 4a2b2c2) × (– 3bc3)= (7 × (– 4) × (–3)) × (a × a2) × (b × b2 × b) × (c2 × c3)
= 84a3b4c5.
Example 2. Find the areas of rectangles with the following pairs of monomials as their lengths
and breadths respectively:
(i)(3x, 4y)
(ii) (9y, 4y2) (iii) (4ab, 5bc) (iv) (2l2m, 3lm2)
Solution.
(i)Area of rectangle = length × breadth
= (3x) × (4y)
= (3 × 4) × x × y = 12xy.
(ii)Area of rectangle = (9y) × (4y2)
= (9 × 4) × y × y2
= 36y3
(iii)Area of rectangle = (4ab) × (5bc)
= (4 × 5) × ab × bc
= 20ab2c