An algebraic expression is a collection of one or more terms, which are separated by the signs + (plus) or / and - (minus).

Some examples of algebraic expressions are $3x, 5x^{2}, 7x +15$

### Value of an Algebraic Expression:

Value of any expression is considered to be the numerical result we get, when any integer will get substituted in place of variables of a given algebraic expression and then simplified.
It may be an expression of multiple values, into which the variables would be given different value and then simplified.

## What is an Algebraic Expression?

An Algebraic expression is an expression that consists of variables and mathematical operations along with numbers. The letter x is the most commonly used variable. The value of an expression depends on the value of its variable(s)

$x, 3x^{2}, 2x +1$ are all Algebraic expressions

### Examples on Algebraic Expressions

Given below are some examples on how to explain the value of an algebraic expression.

Example 1:

If x = 9 and y = 3, find the value of

a) x+y b) x-y c) xy d) $\frac{x}{y}$

Solution:

a) x + y

Substitute the values, x = 9 and y = 3

x + y = 9 +3

= 12

So, the value of the expression x+y is 12

b) x - y

Substitute the values, x = 9 and y = 3

x - y = 9 - 3

= 6

So, the value of the expression x-y is 6

c) xy

Substitute the values, x = 9 and y = 3

xy = 9 x 3

= 27

So, the value of the expression xy is 27

d) $\frac{x}{y}$

Substitute the values, x = 9 and y = 3

$\frac{x}{y}$ = $\frac{9}{3}$

= 3

So, the value of the expression $\frac{x}{y}$ is 3

Example 2:

If p = 8, q =1, r = 2, find the value of :

$\frac{10pq - 3qr}{4pqr - 2p}$

Solution:

Given, p = 8, q = 1, r = 2

10 pq - 3 qr = (10 x 8 x 1) - (3 x 1 x 2)

= 80 - 6

= 74

4pqr - 2p = 4 x 8 x 2 - 2 x 8

= 64 - 16

= 48

$\frac{10pq - 3qr}{4pqr - 2p}$ = $\frac{74}{48}$

= $\frac{37}{24}$

Example 3:

Find the value of the expression, 5x+3, when x = 2, 0, -2.

Solution:

The value of the expression 5x+3 depends on the value of its variable x. That is,

(i) if x = 2, the value of the expression 5x+3 would be,
(5 x 2) + 3 = 10 + 3 = 13

(ii) if x = 0, the value of the expression 5x+3 would be,
(5 x 0) + 3 = 0 + 3 = 3

(iii) if x = -2, the value of the expression 5x+3 would be,
(5 x -2) +3 = -10 + 3 = -7

In order to find the value of an expression, the variable in it is replaced by the given value for the variable and simplified. The method of doing so is called substitution!

## Algebra Terms

The parts of the Algebraic expression that are separated by a (+) sign are called as the terms of the Algebraic expression.

$3x + 4y + z$ is an Algebraic expression. 3x, 4y, z are the terms of the Algebraic expression. All these terms are separated by (+) sign.

$3x^{2} - 2x +7$ is also an Algebraic expression. The terms $3x^{2}$ and $2x$ are separated by (-) sign and not (+) sign. We can rewrite the same expression as $3x^{2} + (-2x) +7$. $3x^{2}$ and $-2x$ are variable terms and 7 is a constant term. Observe that -2x is a term and not 2x.

Algebraic terms with the same factors are said to be like terms. 3x and -8x are like terms. $4xy^{2}$ and $-3xy^{2}$ are like terms. $4xy^{2}$ and $4x^{2}y$ are not like terms as they do not have the same factors.

## Writing an Algebraic Expression

It is often more useful to be able to write an algebraic expression for a real life situation. Given a verbal model that is a verbal description of any situation, it is possible to write a mathematical model.

For example, let us consider a simple situation. There are 1080 apartments in a Resort. Write an expression for the total number of people in the resort if you know the number of people residing in each apartment? Assume each apartment has an equal number of people.

This is a verbal description of a real life situation.

Total number of people = Total number of apartments × Number of people in an apartment

If the total number of people is 'n' and the number of people in each apartment is 'p', then

$n = 1080 \times p$

$n = 1080p$

## Simplifying Algebraic Expressions

Simplifying algebraic expressions means to rewrite the given algebraic expression in a compact form. This is done mainly by adding or subtracting all the terms that can be added or subtracted, removing brackets and so on.

Each term in an algebraic expression is separated by a mathematical sign. The term without any variable is called as a constant. The numerical factor of a variable term is called its numerical coefficient.

Only like terms can be added or subtracted. Like terms are terms with the same variables raised to the same power. For example, 2x + 8x is an algebraic expression with two terms. These two terms are like terms as they have the same variable x and the power of the variable is one in both the terms. Hence, this expression can be simplified by adding up the like terms.

2x + 8x = 10x. This is a single term.

Now consider 3(x + 3y). This is also an algebraic expression which can be simplified by using Distributive Law.

Distributive Law: The sum of any two addends multiplied by a number is equal to the sum of the product of each addend and the number.

For any three variables x, y and z, we have,

x(y+z) = xy + xz

Using the distributive law, we have, 3(x + 3y) = 3x + 9y. The bracket has been removed.

This cannot be simplified further as the two terms are not like terms. One variable is x and the other variable is y. The variables are not same.

We cannot simplify an expression like 3s + $s^{2}$. The two terms have the same variables but their powers are not the same. Hence, they are unlike terms.

In order to simplify an algebraic expression, we must be able to distinguish between like and unlike terms in an expression.

### Properties used to Simplify Algebraic Expression

Given below are some of the properties that are used in simplifying an algebraic expression:

Commutative Property of Addition and Multiplication

a + b = b + a

ab = ba

Associative Property of Addition and Multiplication

a + (b + c) = (a + b) + c

a(bc) = (ab)c

a + 0 = 0 + a = a

Zero is the additive identity element

Distributive Law

a(b + c) = ab + ac

Multiplicative Identity Property

a x 1 = 1 x a = a

One is the Multiplicative identity element

Multiplication by 0

The product of any number and zero is always zero.

a x 0 = 0 x a = 0

a + (-a) = (-a) + a = 0

For every real number 'a', there exists an inverse -a such that the sum of the number a and its inverse -a is zero.

Multiplicative Inverse Property

For any real number 'a' except zero, there is another number $\frac{1}{a}$ (the reciprocal of a) such that a x ($\frac{1}{a}$) = 1

$a$ and $\frac{1}{a}$ are the Multiplicative Inverses of each other.

Division by 1

The quotient of any number divided by 1 is the number itself.

a ÷ 1 = a

Division by 0

Division by 0 is undefined. But, when 0 is divided by any number, the quotient is 0.

$\frac{a}{0}$ = undefined

$\frac{0}{a}$ = 0

With the help of the properties listed above and the steps listed below, we can simplify an algebraic expression efficiently.

Step 1: Simplify the expressions within brackets
Step 2: Simplify the exponents
Step 3: Complete the multiplication and division operations as they occur from left to right.
Step 3: Complete the addition and subtraction operations for like terms as they appear from left to right.

### Examples on Simplifying Algebraic Expressions

Given below are some solved examples on how to simplify algebraic expressions.

Example 1:

Simplify the following

$[\frac{1}{x-1} - \frac{1}{x-3}] / [\frac{1}{x-3} - \frac{1}{x-5}]$

Solution:

Start by simplifying the numerator and the denominator separately.

$\frac{1}{x-1} - \frac{1}{x-3} = \frac{(x-3) - (x-1)}{(x-1)(x-3)}$

$= \frac{-2}{(x-1)(x-3)}$

$\frac{1}{x-3} - \frac{1}{x-5} = \frac{(x-5) - (x-3)}{(x-3)(x-5)}$

$= \frac{-2}{(x-3)(x-5)}$ =

We have the expression reduced to

$\frac{-2}{(x-1)(x-3)} / \frac{-2}{(x-3)(x-5)}$

Using the rule for division of algebraic fractions,

$\frac{-2}{(x-1)(x-3)} / \frac{-2}{(x-3)(x-5)} = \frac{-2}{(x-1)(x-3)} \times \frac{(x-3)(x-5)}{-2}$

$= \frac{(-2)(x-3)(x-5)}{(x-1)(x-3)(-2)}$

$= \frac{(x-5)}{(x-1)}$

Hence the given expression is simplified to $\frac{(x-5)}{(x-1)}$

Example 2:

Simplify $2(3x^{2} + 6x - 1) + 3(5x + 1) - 5x^{2}$

Solution:

Rewrite the expression by using Distributive Property

$2(3x^{2} + 6x - 1) = 2(3x^{2}) + 2(6x) - 2(1)$

$= 6x^{2} + 12x - 2$

$3(5x + 1) = 3(5x) + 3(1)$

$= 15x + 3$

The given expression is now reduced to,

$2(3x^{2} + 6x - 1) + 3(5x + 1) - 5x^{2} = 6x^{2} + 12x - 2 + 15x + 3 - 5x^{2}$

Grouping the like terms and constants together,

$6x^{2} + 12x - 2 + 15x + 3 - 5x^{2} = (6x^{2} - 5x^{2}) + (12x + 15x) + ( -2 + 3)$

Perform the operations that are specified within each of the brackets

$(6x^{2} - 5x^{2}) = x^{2}$

$(12x + 15x) = 27x$

$( -2 + 3) = 1$

Rewrite the expression

$(6x^{2} - 5x^{2}) + (12x + 15x) + ( -2 + 3) = x^{2} + 27x + 1$

There are no like terms. And so, this is the simplest form.

$2(3x^{2} + 6x - 1) + 3(5x + 1) - 5x^{2} = x^{2} + 27x + 1$

## Algebraic Expressions and Polynomials

A polynomial is an expression of finite length constructed from variables and constants. The exponents of the terms are whole numbers. The terms are usually ordered according to degree, either in ascending or descending powers of x. The standard form is where the terms are in descending degrees of x.

e.g. p(x) = $x^{4} + 2x^{2} - x +1$ is a polynomial.

P(x) = $x^{4} + 2x^{-2} -x + 1$ is not a polynomial as the exponent of one of the variables is not a whole number.

p(x) = $a0x^{n} + a1x^{n-1} + a2x^{n-2} + ........ + an$ is the general standard expression of a polynomial. Here a0, a1,....., are called coefficients. a0 is called the leading coefficient. an is called the constant term.

If the value of n is 1, it is called a linear function.

e.g. p(x) = $ax + b$

If the value of n is 2, it is called a quadratic function.

e.g. p (x) = $ax^{2} + bx + c$

If the value of n is 3 it is called a cubic function.

e.g. p (x) = $ax^{2} + bx + cx + d$

Also, the degree of an equation tells how many roots the given polynomial shall have.

If the polynomial is of degree one, then it must have only one zero. If the polynomial is of degree two, then it must have two zeroes.

e.g 2x + 3 = 0 has one zero x = $\frac{-3}{2}$

e.g p(x) = $x^{2} - x - 6 = (x -3 )(x+2)$

Setting p(x) = 0, we get x = 3 or x = -2 ( Two zeroes)

Illustration 1:

State what type of a polynomial is p(x) = $2x6{3} + 3x + 5$.

Here, the value of n = 3.

Hence, it is a cubic polynomial.

Illustration 2:

State what type of a polynomial is $3x^{2}y + 4x - 3y +5$.

Here in $3x^{2}y$, the exponent of x is 2 and the exponent of y is 1. As are multiplied with each other to give 3.

Hence, it is a cubic polynomial.

Illustration 3:

Is p(x) = $x + \frac{1}{x}$ a polynomial?

Here, p(x) = $x + \frac{1}{x}$ can be written as p(x) = x + x^{-1}. Hence, the coefficient of one of the variables is not a whole number.

Hence, it is not a polynomial.