Algebraic Expression

A mathematical expression which consists of numbers, variables and operations is called Algebraic Expression.

Algebraic expressions have no sides (LHS or RHS) or equal to sign ( = ) like algebraic equations.
Examples of algebraic expressions: 5x − 7, 9z² + 4z – 1.

1. Terms

Terms are those parts of an expressions which are separated by an operation ( + or − ). E.g., 3x, 5y and 9 are the terms in the above figure.

2. Factors

Terms might be formed by the product of factors. E.g., term 3x is the product of factors 3 and x.

3. Coefficient

The number placed before the variable or the numerical factor of a term is called Coefficient of that variable. E.g., 3 is the coefficient (numerical factor) of 3x.

4. Variable

A variable can take any value. Its value is not fixed. It is represented by any letter like x, y, p, q, a, b, etc. are called Variables. The variables in the above figure are x and y.

5. Operations

Addition and subtraction are called the operations. These separate each term.

6. Constant

The term which is a number without any variable is called constant. E.g., 9 is constant in above figure.

Number Line and an Expression

An expression can be represented on the number line.

Example

How to represent x + 5 and x – 5 on the number line?

Solution:

First, mark the distance x and then x + 5 will be 5 unit to the right of x.

In the case of x – 5 we will start from the right and move towards the negative side. x – 5 will be 5 units to the left of x.

Monomials, Binomials and Polynomials

Expressions	Meaning	Example
Monomial	The algebraic expression having only one term.	5x2
Binomial	The algebraic expression having two terms.	5x2 + 2y
Trinomial	The algebraic expression having three terms.	5x2 + 2y + 9xy
Polynomial	The algebraic expression having one or more terms with the variable having non−negative integers as an exponent.	5x2 + 2y + 9xy + 4

Binomials, trinomials, … are also polynomials.

Like and Unlike Terms

Terms having the same variable are called Like Terms.

Examples of Like Terms

2x and −9x

24xy and 5yx

6x² and 12x²

The terms having different variable are called, Unlike Terms.

Examples of Unlike Terms

2x and − 9y

24xy and 5pq

6x² and 12y²

Addition and Subtraction of Algebraic Expressions

Steps to add or Subtract Algebraic Expression

First of all, we have to write the algebraic expressions in different rows in such a way that the like terms come in the same column.

Add them as we add other numbers.

If any term of the same variable is not there in another expression then write is as it is in the solution.

Example

Add 15p² – 4p + 5 and 9p – 11

Solution:

Write down the expressions in separate rows with like terms in the same column and add. 

Example

Subtract 5a² – 4b² + 6b – 3 from 7a² – 4ab + 8b² + 5a – 3b.

Solution:

For subtraction also write the expressions in different rows. But to subtract we have to change their signs from negative to positive and vice versa.

Multiplication of Algebraic Expressions

While multiplying we need to take care of some points about the multiplication of like and unlike terms.

1. Multiplication of Like Terms

The coefficients will get multiplied.

The power will not get multiplied but the resultant variable will be the addition of the individual powers.

Example

The product of 4x and 3x will be 12x².

The product of 5x, 3x and 4x will be 60x³.

2. Multiplication of Unlike Terms

The coefficients will get multiplied.

The power will remain the same if the variable is different.

If some of the variables are the same then their powers will be added.

Example

The product of 2p and 3q will be 6pq

The product of 2x²y, 3x and 9 will  be 54x³y

Multiplying a Monomial by a Monomial

1. Multiplying Two Monomials

While multiplying two polynomials the resultant variable will come by

The coefficient of product = Coefficient of the first monomial × Coefficient of the second monomial

The algebraic factor of product = Algebraic factor of the first monomial × Algebraic factor of the second monomial.

Example

25y × 3xy = 125xy²

2. Multiplying Three or More Monomials

While multiplying three or more monomial the criterion will remain the same.

Example

4xy × 5x²y² × 6x³y³ = (4xy × 5x²y²) × 6x³y³

= 20x³y³ × 6x³y³

= 120x³y³ × x³y³

= 120 (x³ × x³) × (y³ × y³)

= 120x⁶ × y⁶

= 120x⁶y⁶

We can do it in other way also

4xy × 5x²y² × 6x³ y³

= (4 × 5 × 6) × (x × x² × x³) × (y × y² × y³)

= 120 x⁶y⁶

Multiplying a Monomial by a Polynomial

1. Multiplying a Monomial by a Binomial

To multiply a monomial with a binomial we have to multiply the monomial with each term of the binomial.

Example

Multiplication of 8 and (x + y) will be (8x + 8y).

Multiplication of 3x and (4y + 7) will be (12xy + 21x).

Multiplication of 7x³ and (2x⁴ + y⁴) will be (14x⁷+ 7x³y⁴).

2. Multiplication of Monomial by a trinomial

This is also the same as above.

Example

Multiplication of 8 and (x + y + z) will be (8x + 8y + 8z).

Multiplication of 4x and (2x + y + z) will be (8x² + 4xy + 4xz).

Multiplication of 7x³ and (2x⁴+ y⁴+ 2) will be (14x⁷ + 7x³y⁴ + 14x³).

Multiplying a Polynomial by a Polynomial

1. Multiplying a Binomial by a Binomial

We use the distributive law of multiplication in this case. Multiply each term of a binomial with every term of another binomial. After multiplying the polynomials we have to look for the like terms and combine them.

Example

Simplify (3a + 4b) × (2a + 3b)

Solution:

(3a + 4b) × (2a + 3b)

= 3a × (2a + 3b) + 4b × (2a + 3b)    [distributive law]

= (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b)

= 6 a² + 9ab + 8ba + 12b²

= 6 a² + 17ab + 12b²     [ba = ab]

2. Multiplying a Binomial by a Trinomial

In this also we have to multiply each term of the binomial with every term of trinomial.

Example

Simplify (p + q) (2p – 3q + r) – (2p – 3q) r.

Solution:

 We have a binomial (p + q) and one trinomial (2p – 3q + r). First multiplying these two.

(p + q) (2p – 3q + r)

= p(2p – 3q + r) + q (2p – 3q + r)

= 2p² – 3pq + pr + 2pq – 3q² + qr

= 2p² – pq – 3q² + qr + pr    (–3pq and 2pq are like terms)

Therefore,

(p + q) (2p – 3q + r) – (2p – 3q) r

= 2p² – pq – 3q² + qr + pr – (2pr – 3qr)

= 2p² – pq – 3q² + qr + pr – 2pr + 3qr

= 2p² – pq – 3q² + (qr + 3qr) + (pr – 2pr)

= 2p² – 3q² – pq + 4qr – pr

Identities

An identity is an equality which is true for every value of the variables.

An equation is not an identity, as an equation is true for only some of the values of the variables.

E.g., x² = 1, is valid if x is 1 but is not true if x is 2. It is an equation but not an identity.

Some of the Standard Identities

(a + b)² = a² + 2ab + b²

(a − b)² = a² – 2ab + b²

a² – b² = (a + b) (a − b)

(x + a) (x + b) = x² + (a + b)x + ab

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

These identities are useful in carrying out squares and products of algebraic expressions. They give alternative methods to calculate products of numbers and so on.