# Chapter 9 – Algebraic Expressions and Identities

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Have algebra terminologies like terms, factors and coefficients bothered you? Ever wished to have an easy run through the whole chapter. Let us try to know what algebraic expressions and identities are all about.

#### What are Expressions?

In Mathematics, an algebraic expression may be defined as a representation or an equation that is made up of variables, constants, and algebraic operations. The latter include any operation like addition, subtraction, multiplication, division, etc. Algebraic expressions are thus made up of terms. For example, 2m + 5n – 10 is an algebraic expression that has the following components:

o   Constants like 2, 5, and 10.
o   Unknown variables m and n.
o   Algebraic operations + and –

There is, however, a basic difference between algebraic expressions and algebraic equations. The former has no equivalent side, meaning that it is not related to another set of algebraic expressions using an = sign, which is the case in algebraic equations.

#### Types of Algebraic Expressions

There are three types of algebraic expressions:

• Monomial Expression or an algebraic expression with only one term like 3x, 4y, 5mn, etc.
• Binomial Expression or an algebraic expression with two terms like 3x + 5y, 5xyz – 25mn, etc.
• Polynomial Expression or an algebraic expression with more than one term and non-negative exponents of a variable like 3x + 5y +9z, or 25t – 12u + 12, etc.

Addition and Subtraction of Algebraic Expressions

Algebra uses the four basic mathematical operations: addition, subtraction, multiplication, and division. Addition and subtraction of algebraic expressions has many similarities with addition and subtraction of numbers. The only difference here is that you must group and sort the like and unlike terms together.

To add two or more algebraic expressions, you must group and sort the like and unlike terms. You need to add the coefficients of the like terms together, using simple addition operation. The common variable is retained as it is. For the unlike terms, you keep them as they are. The resultant sum is represented by another algebraic expression. Let us see how,

Example:

Add 15mn + 2x -12 and 6x – 12mn + 10

First, sort the like terms together,

(15mn – 12mn) + (2x + 6x) – 12 +10

= (15-12) mn + (2+6) x – (12-10) {Applying the rule of BODMAS)

= 3mn + 8x – 2

The final step gives you the required sum.

Subtraction of Algebraic Expressions

Like the addition method, here also you must start by grouping the like and unlike terms together. And then follow the standard subtraction operation.

Example

Subtract 2ab + 15 – 3xy from 6xy + 8ab -20

(2ab+8ab) + (15 – 20) – (3xy + 6xy)

= (2 + 8) ab + {-(20-15)} – (3 + 6) xy

= 10ab + (-5) – 9xy

= 10ab – 9xy -5

The final step gives you the resulting difference.

Multiplication of Algebraic Expressions Introduction

Multiplication of algebraic expressions is a straightforward method and includes two simple steps:

1. Separate the coefficients from the variable.
2. Multiply the coefficients and variables separately to get the product.

Example

Multiply 15m by 2n and 4q

First, separate out the coefficients and multiply them like this,

15 x 2 x 4 = 120

Now, group the variables together like this,

m x n x q = mnq

The required product is 120mnq.

Multiplying A Monomial by A Monomial

The previous example is a case of multiplying a monomial by another monomial. Here is another example:

Find the product of 2x, 3y, 4z

Multiplying the coefficients, you get 2 x 3 x 4 = 24

Grouping the variables together, you get xyz

The required product is 24xyz.

Multiplying A Monomial by A Polynomial

Consider the following example:

Find the product of 3x and (25x+9)

Step 1: Write the expression in the following way:

3x * (25x + 9)

Step 2: Multiply the two terms separately

(3x*25x) + (3x*9)

Step 3: Follow the steps of multiplying a monomial by a monomial

(3*25) (x*x) + (3*9) x

= 75x2 + 27x

The final step gives you the required product.

Multiplying A Polynomial by A Polynomial

Consider the following example:

Multiply (3x + 2y) by (2x2 + 4y)

= 3x (2x2 + 4y) + 2y (2x2 + 4y)

= (3*2) (x*x2) + (3*4) (xy) + (2*2) (yx2) + (y*y)

= 6x3 + 12xy + 4x2y + y2 (this is the required product).

#### What is an Identity?

You all have learnt about algebraic expressions and identities in junior grades. Algebraic identities are nothing but algebraic equations that are valid for all values of the variables. Essentially, you use algebraic identities in computing algebraic expressions.

Standard Identities

The Binomial theorem states that (a+b)n=nC0.an.b0+nC1.an−1.b1+……..+nCn−1.a1.bn−1+nCn.a0.bn

You can derive all the standard identities of algebra from this theorem. Examples of standard identities include:

(a + b)2 = a2 + 2ab + b2

(a-b)2 = a2 – 2ab +b2

a2 – b2 = (a+b) (a-b)

Applying Identities

Example:

Find the product of (m + 9) and (m + 9)

This can be written as (m + 9)2

Applying the standard identity formula, you have a=m and b=9

(m + 9)2 = m2 + 2mn + n2

= m2 + 2m9 + (9)2

= m2 + 18m + 81 (the required product)

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#### FAQs

1. What are algebraic expressions and identities?

Algebraic expressions and identities are representations of a combination of integer constants, variables, and mathematical operations.

2. How many identities are there in algebraic expressions?

There are three basic identities in algebraic expressions:

I. (a+b)^2 = a^2 + b^2 + 2ab
II. (a-b)^2 = a^2 + b^2 – 2ab
III. a^2 – b^2= (a+b)(a-b)

3. What are the identities of algebra?

I. (a+b)^2 = a^2 + b^2 + 2ab
II. (a-b)^2 = a^2 + b^2 – 2ab
III. a^2 – b^2= (a+b)(a-b)

4. What are the 8 algebraic identities?

I. (a+b)^2 = a^2 + b^2 + 2ab
II. (a-b)^2 = a^2 + b^2 – 2ab
III. a^2 – b^2= (a+b)(a-b)
IV. (x + a)(x+b) = x^2 + (a=b)x + ab
V. (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
VI. (a+b)^3 = a^3 + b^3 + 3ab(a+b)
VII. (a-b)^3 = a^3 – b^3 – 3ab(a-b)
VIII. a^3 + b^3 + c^3 – 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)

5. What are the standard identities?

The Binomial theorem states that (a+b)n=nC0.an.b0+nC1.an−1.b1+……..+nCn−1.a1.bn−1+nCn.a0.bn

You can derive all the standard identities of algebra from this theorem.

6. How many types of algebraic expressions are there?

There are three types of algebraic expressions, namely:

• Monomial Expression or an algebraic expression with only one term like 3x, 4y, 5mn, etc.
• Binomial Expression or an algebraic expression with two terms like 3x + 5y, 5xyz – 25mn, etc.
• Polynomial Expression or an algebraic expression with more than one term and non-negative exponents of a variable like 3x + 5y +9z, or 25t – 12u + 12, etc.

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