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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 algebraic expressions are written and know about some basic terms used in writing an algebraic expression.

So an algebraic expression comprises of two parts variable and constant.
Let’s know about them and see how to write an algebraic expression and make addition, subtraction, multiplication and division on them.

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.

Example

5, 25, 100 etc.

These constants and variables when combined with any of these operations \(+\) or \(-\) or \(\times\) or \(\div\) makes an algebraic expression. Let’s see next in details.

Algebraic expression

A combination of constants and variables which are connected by some or all four fundamental operations \(+\), \(-\), \(\times\), \(\div\) is called an algebraic expression.

Example

\(3xy+2\frac{y}{z}+8\frac{x}{z}+9\) is an algebraic expression.
So, here we have made an algebraic expression connected by \(+\) operation by combining variables such as \(3xy, 2\frac{y}{z}, 8\frac{x}{z}\) and a constant 9.

Term

Term is a part in an algebraic expression that comes in between operations \(+\) and \(-\).

In simple words, as we said earlier also, algebraic expression is connected by addition \(+\) and subtraction \(-\).

Therefore, various parts of algebraic expression which are separated by + or – is called term.

Example

Consider an algebraic expression 3xy + 4yz – 8zx.
3xy, 4yz and -8zx are terms of algebraic expression.
Moreover, we can say 3xy + 4yz – 8zx has total of three terms.

Factor

We have learnt above the term and where we can find a term in an algebraic expression.
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.

Example

In this algebraic expression 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.

Example

Consider an algebraic expression 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.

Example

Again in algebraic expression 3xy + 4y + 2xy + 5z
4y and 5z are unlike terms as they have different algebraic factors y in term 4y and z in term 5z.

Operations on algebraic expression

So far, we have learnt the algebraic expression, how it looks like.
Now is the time to learn the real operations on more than one algebraic expressions, similar to those that we do in arithmetic which are the addition, subtraction, multiplication and division.
Let’s go ahead and see how these operations are performed on algebraic expressions with examples.

Addition

How do we add two or more than two algebraic expressions? 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

Example1: Add algebraic expressions
3xy + 7yz
8xy + 5yz

= (3xy + 7yz) + (8xy + 5yz)
Step1: Find like terms in both expressions.
So, the like terms are:
3xy and 8xy
7yz and 5yz
Step2: Add the like terms separately.
= (3xy + 8xy) + (7yz + 5yz)
= (3+8)xy + (7 + 5)yz
= 11xy + 12yz


Example2: Add algebraic expressions
3m2 + 5n2 + 11
7m2 – 2n2 + 6

= 3m2 + 5n2 + 11 + 7m2 – 2n2 + 6
Step1: Find like terms in both expressions.
So, the like terms are:
3m2 and 7m2
5n2 and -2n2
Step2: Add the like terms separately.
= (3m2 + 7m2) + (5n2 + (-2n2)) + (11 + 6)
= 10m2 + (5n2 – 2n2)) + 17
= 10m2 + 3n2 + 17


Example3: Add algebraic expressions
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


Example4: Add algebraic expressions
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

Subtraction operation is similar to addition operation. Again, combine all the like terms and do the subtraction on the numerical coefficients of these like terms.

Example

Example1: Subtract algebraic expressions
3xy + 7yz
8xy + 5yz

= (3xy + 7yz) – (8xy + 5yz)
Step1: Find like terms in both expressions.
So, the like terms are:
3xy and 8xy
7yz and 5yz
Step2: Subtract the like terms separately.
= (3xy – 8xy) + (7yz – 5yz)
= (3 – 8)xy + (7 – 5)yz
= -5xy + 2yz


Example2: Subtract algebraic expressions 3m2 + 5n2 + 11
7m2 – 2n2 + 6

= (3m2 + 5n2 + 11) – (7m2 – 2n2 + 6)
Step1: Find like terms in both expressions.
So, the like terms are:
3m2 and 7m2
5n2 and -2n2
Step2: Subtract the like terms separately.
= (3m2 – 7m2) + (5n2 – (-2n2)) + (11 – 6)
= -4m2 + (5n2 + 2n2) + 5
= -4m2 + 7n2 + 5


Example3: Subtract algebraic expressions
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


Example4: Subtract algebraic expressions 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 addition and subtraction operations of algebraic expressions. In multiplication of algebraic expressions, all terms of first algebraic expressions are multiplied by another algebraic expression.
Let’s learn it by following examples.

Example

Example 1: Multiply algebraic expression 3x + 2y by 4x
= \((3x + 2y) \times 4x\)
Multiply each term of (3x + 2y) to 4x
= \((3x \times 4x) + (2y \times 4x)\)
= \(12x^2 + 8xy\)


Example 2: Multiply algebraic expression 4x2 + 5x + 7 by 8xy
= \((4x^2 + 5x + 7) \times 8xy\)
Multiply each term of \((4x^2 + 5x + 7)\) to 8xy
= \((4x^2 \times 8xy) + (5x \times 8xy) + 7 \times 8xy\)
= \(32x^3 + 40x^2y + 56xy\)


Example 3: Multiply algebraic expression 7x – 3y by 2y + 5z
= \((7x – 3y) \times (2y + 5z)\)
Multiply each term of (7x – 3y) to (2y + 5z)
= \(7x \times (2y + 5z) – 3y \times (2y + 5z)\)
Again, multiply each term of (2y + 5z) with 7x
and multiply each term of (2y + 5z) with 3y
= \(7x \times 2y + 7x \times 5z – 3y \times 2y – 3y \times 5z\)
= \(14xy + 35xz – 6y^2 – 15yz\)


Example 4: Multiply algebraic expression \(\frac{3}{5}a + \frac{7}{2}b\) by \(\frac{7}{2}a\)
= \((\frac{3}{5}a + \frac{7}{2}b) \times \frac{7}{2}a\)
Multiply each term of \((\frac{3}{5}a + \frac{7}{2}b)\) to \(\frac{7}{2}a\)
= \((\frac{3}{5}a + \frac{7}{2}b) \times \frac{7}{2}a\)
= \((\frac{3}{5}a \times \frac{7}{2}a) + (\frac{7}{2}b \times \frac{7}{2}a)\)
= \((\frac{3}{5} \times \frac{7}{2}a^2) + (\frac{7}{2} \times \frac{7}{2}ab)\)
= \(\frac{21}{10}a^2 + \frac{49}{4}ab\)


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 \(\frac{21}{3} = 7\).
We can divide algebraic expressions too the similar way.

Example

Example 1: Divide 12x by 4x
= \(\frac{12x}{4x}\)
= \(\frac{12 \times x}{4 \times x}\)
= 3


Example 2: Divide 24x2y2z2 by 4x2y
= \(\frac{24x^2y^2z^2}{4x^2y}\)
= \(\frac{24 \times x \times x \times y \times y \times z \times z }{4 \times x \times x \times y}\)
= \(6yz^2\)


Example 3: Divide 3x4 + 6x3 – 9x2 + 12x by 3x
= \(\frac{3x^4 + 6x^3 – 9x^2 + 12x}{3x}\)
= \(\frac{3x^4}{3x}\) + \(\frac{6x^3}{3x}\) – \(\frac{9x^2}{3x}\) + \(\frac{12x}{3x}\)
= \(x^3 + 2x^2 -3x +4\)


Example 4: Divide x2 + 5x + 6 by x + 2
= \(\frac{x^2 + 5x + 6}{x + 2}\)
= \(\frac{x^2 + 2x + 3x + 6}{x + 2}\)      (Factorise)
= \(\frac{x(x + 2) + 3(x + 3)}{x + 2}\)
= \(\frac{(x + 3)(x + 2)}{x + 2}\)
= \(x + 3\)