NCERT Class 10 Mathematics Polynomial Expressions CBSE Board Sample Problems (For CBSE, ICSE, IAS, NET, NRA 2022)

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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

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

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

a) Linear polynomial has only one root

b) A zero polynomial has all the real number as roots

c) A constant polynomial has no zeros

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 Theorems

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

Geometric Meaning of the Zero՚s 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 zero՚s 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. noGraph obtainedName of the graphName of the equation
1 where a and b can be any values ()

Example

Quadratic Polynomial
Straight line.

It intersect the x

axis at

Example

Linear polynomial
2

where

and and

Example

Quadratic Polynomial
Parabola

It intersect the x axis at two points

Example

(3,0) and (4,0)

Quadratic polynomial
3

where

and and

Example

Quadratic Polynomial
Parabola

It intersect the x axis at two points

Example

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

Quadratic polynomial
4

where

and and

Example

Quadratic Polynomial
Parabola

It intersect the x axis at one points

Quadratic polynomial
5

where

and and

Example

Quadratic Polynomial
Parabola

It does not intersect the x-axis

It has no zero՚s

Quadratic polynomial
6

where

and and

Example

Quadratic Polynomial
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
It will cut the x-axis shape at the most 3 timesCubic Polynomial
8

Where

It can be of any shape
Polynomial of N Degree
It will cut the x-axis shape at the most n timesPolynomial of n degree

Relation between coefficient and zeroes of the Polynomial

Relation between Coefficient and Zeroes of the Polynomial
S. noType of PolynomialGeneral formZero՚sRelationship between Zero՚s and coefficients
1Linear polynomial1
2Quadratic2

3Cubic3
4

Formation of polynomial when the zeroes are given

Formation of Polynomial when the Zeroes Are Given
Types of polynomialZero՚sPolynomial Formed
Lineark = 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

Steps to divide a polynomial by another polynomial

  • Arrange the term in decreasing order in both the polynomial
  • Divide the highest degree term of the dividend by the highest degree term of the divisor to obtain the first term,
  • Similar steps are followed till we get the reminder whose degree is less than of divisor

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