Class 9 – Algebra

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Algebraic Expression: A combination of literals and numbers connected by one or more of the symbols +, –, ×, and ÷ is called an algebraic expression.

The several parts of an algebraic expression separated by + or – sign are called the terms of the expression.

Example: , , .

Various types of Algebraic Expressions:

i) Monomial: An algebraic expression having only one term is called a monomial.

Example:,……. are monomials.

ii) Binomial: An algebraic expression having two terms is called a binomial.

Example: , , ,…..are binomials.

iii) Trinomials: An algebraic expression having three terms is called a trinomial. Example: , ,……. are trinomials.

iv) Multinomial: An algebraic expression having more than 3 terms is called a multinomial. Example: , , ……. are multinomials.

Polynomial: An algebraic expression in which the powers of variables are all whole numbers is called a polynomial.

Example: , , ……. are polynomials.

But is not a polynomial, as is not a whole number.

General Form: The general form of a polynomial in one variable of degree ‘n’ is as follows

Where n is a whole number and are real numbers.

Degree of a polynomial: The degree of a polynomial in one variable is the highest power of the variable and for a polynomial with multiple variables is the highest sum of all the powers in a single term.

Example:

i) Degree of

ii) Degree of is 4

Operations on polynomials:

Addition: To add two or more polynomials, group the like terms and add their constant coefficients.

The sum of two polynomials is again apolynomial.

Example: Add

Subtraction: To subtract a polynomial from another,change the sign of each term in the expression to be subtracted and then add the two polynomials.

Example: Subtract from

Multiplication: To multiply two or more polynomials multiply the numerical co-efficient and variables separately and write alltogether.

i) Multiply 4ab and -3a2b.

ii) Multiply

Division: If p(x) is a polynomial in x andq(x) is another polynomial in x then the product of these two polynomials is another polynomial say g(x) then we can write

Example:

i) Divide by

ii) Divide by

Identity: An equation that is true for all valuesof the variables is called an Identity.

Example: is true for all values of ‘x’.Here are a few identities listed below.

Important Identities:

i)

ii)

iii)

iv)

v)

vi)

vii)

viii)

ix)

x)

xi)

xii)

xiii)

1. Expand .

Solution:

2. Simplify by using anidentity.

Solution:

We know that

3. If then find .

Solution:

4. Expand

Solution:

We know that

5. If then show that

Solution: (given)

We know that

Hence proved.

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We know the identity: …(1)

Consider

From (1) and (2)