**ALGEBRA**

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and my dear students! In this post, I have covered **class -7 Introduction
to ****Algebra (**Foundation** IIT)**** ****important points** After Reading **Mathematics ****Algebra ****(7th class ) ****IIT**** Notes With important points**, please do share it with your friends.
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Variable: A letter symbol which can take any value of a certain set.

Eg: a, x , y., etc.,

Constants: Quantities which have only one
fixed value.

Eg : 2, 5, sometimes letters are indicated as
constants.

Term: Numerical numbers alone or
literal numbers alone or their combinations by operation of multiplication. Eg:
2a, 3x.

Exponent & Base: If a¹0 then a x a x a.... m times = a^{m}, here ‘a’ is called base and
‘m’ is called exponent. ‘a^{m }’ can be read as m^{th} power of
‘a’.

Eg: In x^{2}, x is base and 2 is exponent or index

Algebraic
Expression: A
combination of terms by the Mathematical operations. Eg: 3x–2y, x+2, 5x^{2 }

Coefficient: In a product containing two or more than two factors, each
factor is called the coefficient of the product of other factors.

Eg: In 2a, 2 is coefficient of ‘a’ which is
called as numerical coefficient. ‘a’ is called literal coefficient of 2.

Like and Unlike Terms : Terms which contain the same
variable or literal factor are called like terms, othe rwise they are called
unlike terms.

Eg: x^{2},5x^{2},7x^{2} are like terms. 5x,7 x^{2},3y are unlike terms.

Value of an Algebraic
Expression: By
substituting numerical values in place of variables gives the value of
Algebraic expression. Eg: In 3x+4, when x=2 then 3x+4=6+4=10 Types of

Algebraic expressions:

1. Monomial:
Having one term. Eg: 2x, 3y^{2}, 5

2. Binomial :
Having two terms. Eg: 3x+2y, x^{2}+2

3. Trinomial:
Having three terms. Eg: a+b+c

4. Multinomial :
Having more than three terms Eg: ax + by + c + 2

Polynomial: An algebraic expression in
which the variables involved have only non – negative integral powers, is
called a polynomial.

Eg : 1.
2/ 3 x^{ 3}- 3 /4x^{2} is a polynomial in variable x where a 1/2 x^{
3} -3x^{2}+5x^{1/2}+x-1^{ }is not a polynomial,
because it contains a term 5 x ^{1/2} which contains 1/2 as the power of
variable x, which is not a non – negative integer.

2. 1 +
2x + 3y, 4x^{2}y – 5y etc., are polynomials No polynomial should
contain a term having a variable in its denominator

Eg: x +
1/x , x^{2}+2x+1/x^{2 }are
not

Polynomial in one
variable: An
algebraic expression involving only one variable in which the powers of the
variable are non – negative integers, is a polynomial in that variable.

Eg : 3x + 7 is a polynomial in x,

2y^{2} – 5y + 1 is a polynomial in
y

Polynomials in two or more variables: An

algebraic
expression involving two or more variable with non – negative integral powers
is called a polynomial in these variables.

Eg : x + y + xy is a polynomial in x & y,

a^{2 }+
ab^{2} + 3ab + 5 is a polynomial in a & b

Difference between multinomial and polynomial:

1. A polynomial is concerned with positive integral powers of the variable involved
in the expression and not with the number of terms in the expression.

2.A
multinomial is concerned with number of terms in the expression and not with
the powers of the variable in the expression. Thus

all
polynomials are multinomials, But all

multinomials
need not be polynomials.

Eg:3√ x+4/xis a multinomial but not a polynomial

More definitions: If the coefficients of the terms in a
polynomial are integers, it is called a polynomial with integral coefficient.
Eg : z ^{2} + 4z^{2} + 3z – 6, here coefficients of terms 1, 4,
3, -6 are integers If the coefficients of the terms in a polynomial are
rationals, it is called a polynomial with rational

1/3 x^{2}y+4/5xy^{2}+xyz here coefficients of

Terms 1/3,4/5,1are rationals.

**Also Check**

**Introduction to Knowing Our Numbers Key Points**

**Introduction to integers (7th class)**

## Introduction to Real number

**Trigonometry Do This & Try this solution**

**SSC (10th class) Trigonometry Exercise - 11.1 Solution**

**SSC(10th class) Trigonometry Exercise - 11.1 Solutions**

An algebraic expression of the form
a + bx + cx^{2} +……….is called a polynomial in a single variable x. Here a, b, c ….. are rationals such that atleast one of them is non – zero.
If all the coefficients a, b, c, are zero then it is called a zero polynomial.
Since a, b, c…. are zeros then polynomial is 0 + 0. x + 0. x^{2} +…..= 0. Hence 0 is called zero polynomial.

Zero of a polynomial: The number for which the value of a polynomial is zero, is called zero of the polynomial.

Eg: Find the zero of the polynomial ax + b

Sol: ax
+ b = 0

ax = -b

ax = -b

x = -b/a

So zero of the polynomial ax + b is -b/a

Difference
between zero polynomial and

zero of
the polynomial :

Zero
polynomial is one type of polynomial where as zero of a

polynomial is a number at which the value of the polynomial is
zero.

Degree of a term in a polynomial : The degree of any term of a polynomial is the sum of the

powers of all variables in that term.

Eg : 1. In 3x^{2}y^{4}z^{6}, Only term is 3x^{2}y^{4}z^{6}

Its degree is 2 + 4 + 6 =12

Eg : 2. In x + y + xy, terms are x, y,. xy

Their degrees are1, 1, 2

Eg :3. In 5x^{3} + 3x^{2} + 4x – 8, terms are 5x^{3}, 3x^{2},4x and - 8 Their
degree is 3, 2, 1, 0

Note that:
Degree of a constant term is zero

since it involves no variable

Eg : 5 = 5. x^{0} so degree of 5 is 0.

Degree of the polynomial : The degree of the highest degree term in a polynomial is called the Degree of the polynomial.

Eg :

1. Degree of x + 2y + x y is 2

2. Degree of 3x^{2}y^{4}z^{6 }is 12

3. Degree of 5x^{3} + 3x^{2} + 4x – 8 is 3

4. Degree of a^{2}b + ab^{2}
+ 3ab + 5 is 3

5. Degree of y + z is 1

1. Degree of zero polynomial is not defined

2. Degree of a non zero constant polynomial is zero

Linear polynomial: A polynomial of degree‘1’ is called a linear
polynomial.

Eg : 3+5x

Linear polynomial.General form of linear polynomial in

variable ‘x’ is a x + b (a ≠0)

Quadratic polynomial:A polynomial of degree ‘2’ is
called Quadratic polynomial.5x^{2} + 3x+ 4 is ax^{2} + bx+ c (a≠0)

Quadratic polynomial.General form of quadratic polynomial in

Variable ‘x’ is ax^{2}
+ bx+ c (a≠0)

Cubic Polynomial :A polynomial of degree‘3’ is called cubic polynomial

5x^{3}+ 3x^{2}+ 4x+1 is
a Cubic polynomial

General form of a cubic polynomial in

variable x, ax3+ bx2+ cx+ d (a≠0)

Biquadratic polynomial:A polynomial of degree ‘4’ is called Bi Quadratic polynomial

Eg: 5x^{4}+ 5x^{3}+ 3x^{2}+ 4x+1 are bi
quadratic polynomials

General Form of a Bi Quadratic polynomials in variable ‘x’ is ax^{4}+bx^{3}+cx^{2}+dx+e
(a≠0)

n^{th }degree polynomial :ax^{n} + bx^{n-1}+ cx^{n-2} +…+ px + q(a≠0) is
called a polynomial of degree ‘n’ in one variable in x

Where a, b, c, ……… p, q are complex
numbers.

[Here the numerical coefficient of
highest degree term is called leading coefficient of that polynomial]

Formulae :

1. (a + b) ^{2 }= a^{2}
+ 2ab + b^{2}

2. (a - b) ^{2 }=
a^{2} - 2ab + b^{2}

3. (a + b) (a - b) = a^{2} - b^{2}

4. (a + b)3 = a3 + 3a2b + 3a b^{2}
+ b3 (or)

a3 + 3ab (a + b ) + b3

5. (a- b)3 = a3 - 3 a^{2}b
+ 3a b^{2} - b3 (or)

= a3 - 3ab (a - b) - b3

6. (a+ b+ c) 2 = a^{2}+ b^{2} + c2 + 2ab
+ 2bc + 2ca

7. a3 + b3 = (a + b) (a2- ab + b^{2} )

8. a3 - b3 = (a - b) ( a^{2} + ab
+ b^{2} )

9. (a + b)2 - (a- b)2 = 4ab

10. (a + b)2 + (a- b)2 = 2a^{2} +
2^{2}

(Or)2( a^{2} + b^{2} )

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