Revision Notes on Algebraic Expressions and Identities

Algebraic Expression

Any mathematical expression which consists of numbers, variables and operations are called Algebraic Expression.

1. Terms

Every expression is separated by an operation which is called Terms. Like 7n and 2 are the two terms in the above figure.

2. Factors

Every term is formed by the product of the factors.7n is the product of 7 and n which are the factors of 7n.

3. Coefficient

The number placed before the variable or the numerical factor of the term is called Coefficient of that variable.7 is the numerical factor of 7n so 7 is coefficient here.

4. Variable

Any letter like x, y etc. are called Variables. The variable in the above figure is n.

5. Operations

Addition, subtraction etc. are the operations which separate each term.

6. Constant

The number without any variable is constant. 2 is constant here.

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

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     [Since 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)

(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)

(2p – 3q) r = 2pr – 3qr

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 variable but an equation is true for only some of the values of the variables.

So an equation is not an identity.

Like, x² = 1, is valid if x is 1 but is not true if x is 2.so 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.

Applying Identities

Example

(4x – 3y)²

= (4x)² – 2(4x) (3y) + (3y)² 

= 16x² – 24xy + 9y²

Example

Use the Identity (x + a) (x + b) = x² + (a + b) x + ab to find the value of 501 × 502

Solution:

 501 × 502

= (500 + 1) × (500 + 2)

 = 500² + (1 + 2) × 500 + 1 × 2

= 250000 + 1500 + 2

= 251502