2.1 Algebra
Algebra is a branch of mathematics which deals with
known values substituted for unknowns
Do you have interest in finding solutions to
problems similar to:
1. Find 9.5*10.5 quickly
2. If the difference
between 2 numbers is 8 and the difference between their squares is 400, What
are those numbers? (Lilavati Shloka 59)
3. Sum of my age and my
father’s age is 55 years. If after 16 years, my father’s age is twice as that
of mine, then can you tell me my age as on today?
4.
Suppose you along with your friends had planned a picnic.
You had budgeted Rs.480 for food. But at the last moment 8 of your friends did
not go for the picnic. Because of their absence other members paid Rs.10 extra
for food. Find out how much each one paid finally?
5.
A plane left 30 minutes later than the scheduled time. In
order to reach the destination 1500 km away it has to increase the speed by
250km/hr from its regular speed. Find the regular speed and its normal journey
time.
We are solving these problems with the help of
algebra in 2.4,
2.8, 2.14, 2.19
respectively.
2.1 Introduction to Algebra:
A Constant is
a number whose value does not change. Examples are 4,0,1/3,5/2,1.19,_{}
A Variable is
the one which does not have a fixed value and can take any value. Variables are
represented by alphabets. The examples are x, y, a+b. They may take numerical
values depending upon the problem
An Algebraic term
is a number, a variable or the product of number and variables.
Examples are 4ab, 2x, 3y, 10, z, m/n, p/q…
Let us look at the term 2x= 2*x which is the
product of 2 and x. We call 2 as the numerical
coefficient of 2x. Similarly x is called literal factor or literal coefficient of 2x.Terms having same
literal factors with same exponents are called like
terms.
Examples are
( x,2x, 7x) >(These have x as same
literal factor)
(2mn, 5mn,1/3mn) >( These have mn as same literal
factor)
(x^{3}, 5 x^{3}, 5/6 x^{3}) >(These have x^{3} as the same literal
factor). In these examples 3 is exponent (power) of x
Terms having different literal factors or same
literal factors with different exponents are called unlike
terms
Examples are:
( x, x^{3}, x^{2}) >(Though all have same x
as literal factor the exponents are 1, 2 and 3 which are not same)
( 2x, 2a,2mn) >( These have x, a and mn as different literal
factors)
An algebraic expression is a combination of
constants and variables. (We can say it is combination of like and or unlike
terms). Examples are
4x+ax^{3}+9x^{2}+ (2a/3b), 2mn+45+
y^{2}+ _{} +_{}
A ‘polynomial’ is an algebraic expression in which variables
have only positive integral exponents. Examples are:
4x+ax^{3}+9x^{2}+(2a/3b), 2mn+45
Note y^{2}+
x^{3/2 } is not a polynomial because y has – 2 as exponent and x
has 3/2 as exponent which are not positive
integers.

Types
of Algebraic expression 
Examples 
An
algebraic expression is called 
monomial if it has one
term 
3a,
2,1/3y, 
An
algebraic expression is called 
binomial if it has two terms 
34a,
5x^{2}z 
An
algebraic expression is called 
trinomial if it has three terms 
4x+ax^{3}+9x^{2} 
A polynomial is said to be in standard form if its terms are in ascending/descending
powers of the variables:
Example :
The term y^{2}2y^{4}+3yy^{3}+4
is in non standard form.
Same term is rewritten in the standard form as
2y^{4}y^{3}+ y^{2}+3y+4 or 4+3y+ y^{2}y^{3}2y^{4}
If the number of variables in the polynomial is ‘n’
then it is called a polynomial in ‘n’ variables.
Examples :
1. 9x^{5}+3x^{3}+9x^{2}+7x+5
is polynomial in one variable(x is the only variable)
2. 9x^{5}+ax^{3}+9x^{2}+7x+5
is polynomial in two variables(x and a are the 2 variables)
The highest of sum of exponents of all variables in a polynomial is called the degree of the polynomial.
Note: If p(x) is a polynomial of degree m and q(x) is a polynomial of
degree n then the product p(x)*q(x) is a polynomial of degree m+n.
Examples:
1. The degree of 4y^{2} x^{2}y^{2}+
x^{2}+6y is 4 (_{} the sum of exponents of x^{2}y^{2} is
4(=2+2) which is the highest compared to the exponents of other terms 4y^{2},
x^{2}, 6y which are 2, 2 and
1 respectively)
2. The degree of 10p^{3}q^{2}+4p^{2}q5+_{}p^{4} is 5 (_{} the sum of exponents of p^{3}q^{2} is
5(=3+2) which is the highest compared to the exponents of other terms 4p^{2}q
,_{}p^{4} which
are 3(=2+1) and 4 ^{ }respectively)
The rules and properties (associative,
distributive) applicable for arithmetic operations also apply for arithmetic
operations involved with variables
For example we know 5 – (6) = 5+6 =11: 2 – (+5) = 25 = 7 and so on. Similarly
(a+b)+c =a+(b+c) …….
In the case of algebraic expressions, while adding
like terms, their numerical coefficients are added:
For example:
1)8y^{4 }2y^{4}=(82)y^{4}=6y^{4}
2) 11ab +6ab = {11+(6)}ab = (116)ab = 17ab
Unlike terms can not be added.
For example 8y^{4 }2y^{2} can not
be further added and simplified
2.1.Problem 1: Add 5a^{2}6a+3, 2a^{2}+3a1, 3a^{2}a5
Solution :
The problem can be
written as
(5a^{2}6a+3)+ (2a^{2}+3a1) + (3a^{2}a5)
=5a^{2}6a+3+2a^{2}+3a1+3a^{2}a5
= (5a^{2}+2a^{2} +3a^{2}) +
(6a+3aa) + (315) (By grouping like terms to be together)
= (5+2+3) a^{2 }+ (6+31)a + (315) (By
adding the co efficients of like terms)
=10 a^{2 }+ (4a)3
=10a^{2} 4a3
2.1.Problem 2: Subtract 2x^{3}x^{2}+4x6 from x^{3}+5x^{2}4x+6
Solution :
The problem can be
written as
(x^{3}+5x^{2}4x+6) – (2x^{3}x^{2}+4x6)
= x^{3}+5x^{2}4x+6  2x^{3 }(x^{2})
(+4x) –(6)
=x^{3}+5x^{2}4x+6  2x^{3}+x^{2}4x+6
(_{} ( x^{2}) = x^{2} and –(6) =+6 )
= (x^{3}  2x^{3})+(5x^{2}+x^{2})+(4x
4x) +(6+6) (By grouping like terms to be together)
=  x^{3}+6x^{2}8x +12
2.1.Problem 3 : What must be subtracted from x^{3}+2x^{2}3x+7
to get x^{3}+x^{2}+x 1
?
Solution :
The problem is similar to ‘what must be subtracted
from 9 to get 3? The answer we know is 6 and is got by 93(=6)
Similarly we need to find (x^{3}+2x^{2}3x+7)
– (x^{3}+x^{2}+x 1)
(x^{3}+2x^{2}3x+7) – (x^{3}+x^{2}+x
1)
= x^{3}+2x^{2}3x+7 – x^{3}x^{2}x
–(1)
= (x^{3}– x^{3})+(2x^{2}x^{2})+(3x
–x) +(7+1) (_{} –(1) =+1 )
= 0+x^{2}4x+8
= x^{2}4x+8
In this example x^{3}+2x^{2}3x+7
is minuend (x^{3}+x^{2}+x 1) is subtrahend and
x^{2}4x+8 is the difference
Verification:
We know that
Minuend = subtrahend + difference
(x^{3}+x^{2}+x 1) + (x^{2}4x+8)
= x^{3}+(x^{2} + x^{2})
+(x4x) 1+8 (By grouping like terms to be together)
= x^{3}+2x^{2}3x 7 which is
minuend given in the problem
2.1 Summary of learning
No 
Points studied 
1 
Definitions
of Constants, variables, Degree, Monomial,
Binomial, Trinomial, Polynomial 