NCERT Class 10 Mathematics Formula CBSE Board Sample Problems Part 2

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Polynomial Expressions

A polynomial expression S(x) in one variable x is an algebraic expression in x term as

Where are constant and real numbers and is not equal to zero

Some Important Point to Note

Some Important point to Note

S.no

Points

1

are called the coefficients for

2

n is called the degree of the polynomial

3

When all are zero, It Is called zero polynomial

4

A constant polynomial is the polynomial with zero degree. It is a constant value polynomial

5

A polynomial of one Item is called monomial, two Items binomial and three Items as trinomial

6

A polynomial of one degree is called linear polynomial, two degree as quadratic polynomial and degree three as cubic polynomial

Important Concepts on Polynomial

Important concepts on Polynomial

Concept

Description

Zero’s or roots of the polynomial

It is a solution to the polynomial equation S(x) =0 i.e. a number “a” is said to be a zero of a polynomial if S (a) = 0.

If we draw the graph of S(x) =0, the values where the curve cuts the X-axis are called Zeros of the polynomial

Remainder Theorem’s

If p(x) is an polynomial of degree greater than or equal to 1 and p(x) is divided by the expression (x-a), then the remainder will be p(a)

Factor’s Theorem’s

If x-a is a factor of polynomial p(x) then p(a)=0 or if p(a)=0,x-a is the factor the polynomial p(x)

Geometric Meaning of the Zeroes of the Polynomial

Let’s us assume

y= p(x) where p(x) is the polynomial of any form.

Now we can plot the equation y=p(x) on the Cartesian plane by taking various values of x and y obtained by putting the values. The plot or graph obtained can be of any shapes

The zeroes of the polynomial are the points where the graph meet x-axis in the Cartesian plane. If the graph does not meet x-axis, then the polynomial does not have any zero’s.

Let us take some useful polynomial and shapes obtained on the Cartesian plane

Geometric Meaning of the Zeroes of the polynomial

S.no

Graph obtained

Name of the graph

Name of the equation

1

where a and b can be any values ()

Example

Quadratic polynomial

Quadratic Polynomial

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Straight line.

It intersect the x

axis at

Example

Linear polynomial

2

where

and and

Example

Quadratic polynomial

Quadratic Polynomial

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Parabola

It intersect the x axis at two points

Example

(3,0) and (4,0)

Quadratic polynomial

3

where

and and

Example

Quadratic polynomial

Quadratic Polynomial

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Parabola

It intersect the x axis at two points

Example

(-2,0) and (4,0)

Quadratic polynomial

4

where

and and

Example

Quadratic polynomial

Quadratic Polynomial

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Parabola

It intersect the x axis at one points

Quadratic polynomial

5

where

and and

Example

Quadratic polynomial

Quadratic Polynomial

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Parabola

It does not intersect the x-axis

It has no zero’s

Quadratic polynomial

6

where

and and

Example

Quadratic polynomial

Quadratic Polynomial

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Parabola

It does not intersect the x-axis

It has no zero’s

Quadratic polynomial

7

Where

It can be of any shape

Cubic Polynomial

Cubic Polynomial

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It will cut the x-axis shape at the most 3 times

Cubic Polynomial

8

Where

It can be of any shape

Polynomial of n degree

Polynomial of N Degree

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It will cut the x-axis shape at the most n times

Polynomial of n degree

Relation between Coefficient and Zeroes of the Polynomial

Relation between coefficient and zeroes of the Polynomial

S.no

Type of Polynomial

General form

Zero’s

Relationship between Zero’s and coefficients

1

Linear polynomial

1

2

Quadratic

2

3

Cubic

3

4

Formation of Polynomial when the Zeroes Are Given

Formation of polynomial when the zeroes are given

Types of polynomial

Zero’s

Polynomial Formed

Linear

k=a

(x-a)

Quadratic

and

Or

Or

Sum of the zero’s)x +product of the zero’s

Cubic

and

Division Algorithm for Polynomial

Let’s p(x) and q(x) are any two polynomial with ,then we can find polynomial s(x) and r(x) such that

Where r(x) can be zero or degree of degree of g(x)

Dividend