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 