# 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

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

 Types of polynomial Zero՚s Polynomial Formed Linear k = a (x-a) Quadratic and OrOr 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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