Introduction
As we all know, algebra is a branch of mathematics where we study variables and constants and learn operations on them, such as addition, subtraction, multiplication and division, the similar operations that are used in arithmetic. In algebra operations are performed on algebraic expressions whereas in arithmetic, operations are performed on only numbers.
Before getting a deep dive into algebraic operations, first we understand what are the parts of an algebraic
expression, how it is written and know some of its basic terms.
So, it comprises of two parts variable and constant.
Variable
A symbol which can take different numerical value is called a variable. We use any letters such as a, b, c, x, y, z etc. to represent variable.
Constant
Constant which has a fixed numerical value.
5, 25, 100 etc.
These constants and variables when combined with any of these operations + or – or × or ÷ result to an algebraic expression.
Algebraic expression
A combination of constants and variables which are connected by some or all four fundamental operations +, -, ×, ÷ is called an algebraic expression.
3xy +
2y
z
+
8x
z
+ 9
This example is of an algebraic expression connected by + operation by combining variables
3xy,
2y
z
,
8x
z
and a constant 9.
Term
Term is a part in an algebraic expression that comes in between operations + and -.
In other words, the various parts of algebraic expression which are separated by + or – are called terms.
3xy + 4yz – 8zx
3xy, 4yz and -8zx are terms
Moreover, we can say 3xy + 4yz – 8zx has total of three terms.
Factor
We have learnt above the term and how to find it in algebraic expressions.
So, here comes the factors. As we know terms are formed by the product of variables and constants, so these
variables and constants in a term are known as factors.
So, a term can be written as product of factors.
These factors contain a numerical coefficient and an algebraic factor.
In 3xy + 4yz – 8zx, 3 is numerical factor and xy is algebraic factor.
4 is numerical factor and yz is algebraic factor.
-8 is numerical factor and zx is algebraic factor.
Further, we can classify terms on the basis of factors into Like Terms and Unlike Terms. Let’s see how next.
Like Terms
When terms have same algebraic factor, then they are called as like terms.
3xy + 4y + 2xy + 5z
Here, 3xy and 2xy are like terms as they have same algebraic factors x and y.
Unlike Terms
When terms have different algebraic factor, then they are called as unlike terms.
3xy + 4y + 2xy + 5z
Here, 4y and 5z are unlike terms as they have different algebraic factors y in term 4y and z in term 5z.
Let’s see next how to do operations of addition, subtraction, multiplication and division on them.
Operations on algebraic expression
Operations on algebraic expressions are similar to the arithmetic operations like addition, subtraction,
multiplication and division. Let’s take a look at how these operations are performed with examples.
Addition
How are two or more than two algebraic expressions added?
It is simple, just combine all the like terms and do the addition on the numerical coefficients of these like terms.
So, rule of thumb is combine only like terms and the usual addition operation on coefficients of all like
terms.
Example 1: Add
3xy + 7yz
8xy + 5yz
= (3xy + 7yz) + (8xy + 5yz)
Step 1: Find like terms in both expressions.
So, the like terms are:
3xy and 8xy
7yz and 5yz
Step 2: Add the like terms separately.
= (3xy + 8xy) + (7yz + 5yz)
= (3+8)xy + (7 + 5)yz
= 11xy + 12yz
Example 2: Add
3m2 + 5n2 + 11
7m2 – 2n2 + 6
= 3m2 + 5n2 + 11 + 7m2 – 2n2 + 6
Step 1: Find like terms in both expressions.
So, the like terms are:
3m2 and 7m2
5n2 and -2n2
Step 2: Add the like terms separately.
= (3m2 + 7m2) + (5n2 + (-2n2)) + (11 + 6)
= 10m2 + (5n2 – 2n2) + 17
= 10m2 + 3n2 + 17
Example 3: Add
8x2 + 11y2 + 6z2
9x2 – 2y2 – 7z2
2x2 + 4y2 + 5z2
= (8x2 + 11y2 + 6z2)
+ (9x2 – 2y2 – 7z2)
+ (2x2 + 4y2 + 5z2)
Combine like terms.
= (8x2 + 9x2 + 2x2)
+ (11y2 – 2y2 + 4y2)
+ (6z2 – 7z2 + 5z2)
= 19x2 + 13y2 + 4z2
Example 4: Add
20a + 12 and
7b + 8
= (20a + 12) + (7b + 8)
We usually combined like terms in the above examples, but we here unlike terms also exist.
Here, 20a and 7b are unlike terms.
= 20a + 7b + (12 + 8)
= 20a + 7b + 20
Subtraction
The subtraction operation is similar to the addition operation. To subtract algebraic expressions, combine all the like terms and do the subtraction on the numerical coefficients of these like terms.
Example 1: Subtract
3xy + 7yz
8xy + 5yz
= (3xy + 7yz) – (8xy + 5yz)
Step 1: Find like terms in both expressions.
So, the like terms are:
3xy and 8xy
7yz and 5yz
Step 2: Subtract the like terms separately.
= (3xy – 8xy) + (7yz – 5yz)
= (3 – 8)xy + (7 – 5)yz
= -5xy + 2yz
Example 2: Subtract
3m2 + 5n2 + 11
7m2 – 2n2 + 6
= (3m2 + 5n2 + 11) – (7m2 – 2n2 + 6)
Step 1: Find like terms in both expressions.
So, the like terms are:
3m2 and 7m2
5n2 and -2n2
Step 2: Subtract the like terms separately.
= (3m2 – 7m2) + (5n2 – (-2n2)) + (11 – 6)
= -4m2 + (5n2 + 2n2) + 5
= -4m2 + 7n2 + 5
Example 3: Subtract
8x2 + 11y2 + 6z2
9x2 – 2y2 – 7z2
= (8x2 + 11y2 + 6z2)
– (9x2 – 2y2 – 7z2)
Combine like terms.
= (8x2 – 9x2)
+ (11y2 – (-2y2))
+ (6z2 – (-7z2))
= -x2 + (11y2 + 2y2) + (6z2 + 7z2)
= -x2 + 13y2 + 13z2
Example 4: Subtract 7b + 8 from 20a + 12
= (20a + 12) – (7b + 8)
here we have unlike terms.
20a and 7b are unlike terms.
= 20a – 7b + (12 – 8)
= 20a – 7b + 4
Multiplication
We can also multiply the algebraic expressions. Here we do not need to combine like terms as we did above in
the addition and subtraction operations. In multiplication, all
terms of first expression are multiplied by the terms of second expression.
Let’s learn it by following examples.
Example 1: Multiply 3x + 2y by 4x
= (3x + 2y) × 4x
Multiply each term of (3x + 2y) to 4x
= (3x × 4x) + (2y × 4x)
= 12x2 + 8xy
Example 2: Multiply 4x2 + 5x + 7 by 8xy
= (4x2 + 5x + 7 × 8xy
Multiply each term of (4x2 + 5x + 7) to 8xy
= (4x2 × 8xy) + (5x × 8xy) + 7 × 8xy
= 32x3 + 40x2y + 56xy
Example 3: Multiply 7x – 3y by 2y + 5z
= (7x – 3y) × (2y + 5z)
Multiply each term of (7x – 3y) to (2y + 5z)
= 7x × (2y + 5z) – 3y × (2y + 5z)
Again, multiply each term of (2y + 5z) with 7x
and multiply each term of (2y + 5z) with 3y
= 7x × 2y + 7x × 5z – 3y × 2y – 3y × 5z
= 14xy + 35xz – 6y2 – 15yz
Example 4: Multiply
3a
5
+
7b
2
by
7a
2
=
(
3a
5
+
7b
2
)
×
7a
2
Multiply each term of
(
3a
5
+
7b
2
)
to
7a
2
=
(
3a
5
×
7a
2
) +
(
7b
2
×
7a
2
)
=
(
3
5
×
7
2
a2) +
(
7
2
×
7
2
ab)
=
21a2
10
+
49ab
4
Division
Divisions involved in algebraic expressions are all similar to that we do in arithmetic. In arithmetic, we know
that when 21 is divided by 3, we get number 7 which can be written as
21
3
= 7.
We can divide algebraic expressions in the similar way too.
Example 1: Divide 12x by 4x
=
12x
4x
=
12 × x
4 × x
= 3
Example 2: Divide 24x2y2z2 by 4x2y
=
24x2y2z2
4x2y
=
24 × x × x × y × y × z × z
4 × x × x × y
= 6yz2
Example 3: Divide 3x4 + 6x3 – 9x2 + 12x by 3x
=
3x4 + 6x3 – 9x2 + 12x
3x
=
3x4
3x
+
6x3
3x
–
9x2
3x
+
12x
3x
= x3 + 2x2 – 3x + 4
Example 4: Divide x2 + 5x + 6 by x + 2
=
x2 + 5x + 6
x + 2
=
x2 + 2x + 3x + 6
x + 2
(Factorise)
=
x(x + 2) + 3(x + 3)
x + 2
=
(x + 3)(x + 2)
x + 2
= x + 3
