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Polynomials

Polynomial definition: Let x be a variable (literal), n be a positive

integer and ��, ��, ��, ��…�be constants (real numbers).

Then� + ����� + ����� +⋯�� + �� is known as a

polynomial in a variable x.

Degree of a polynomial: The exponent of the highest degree term in

a polynomial is known as its degree.

Constant polynomial: A polynomial of degree zero is known as

constant polynomial. E.g. ��

Linear polynomial: A polynomial of degree one is known as linear

polynomial e.g. � + �

Quadratic polynomial: A polynomial of degree two is known as

quadratic polynomial e.g. �� + � + �

Cubic polynomial: A polynomial of degree two is known as cubic

polynomial e.g. �� + �� + � + �

Bi-quadratic polynomial: A polynomial of degree four is known as bi-

quadratic polynomial e.g. �� + � + �

Value of a polynomial: The value of a polynomial �() at = � is

obtained by substituting = � in the given polynomial and is

denoted by�(�).

Zero or root of a polynomial: A real number � is root or zero of

polynomial �() = � + ����� + ����� +⋯�� + �� if

�(�) = 0 i.e. �(�) = �� + ������ + ������ +⋯��� +

�� = 0.

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Some important facts:

1. Integral Root Theorem: If �() is polynomial with integral

coefficients and the leading coefficient is 1, then any integer

root of �() is a factor of the constant term.

E.g. �() = � + 2� − 11 − 12 has an integer root, then it is

one of the factors of 12 which are

±1,±2,±3,±4,±6,±12 . In fact 3 is a root of a polynomial

as�(3) = 0.

2. Rational root Theorem: Let �

� be a rational fraction in its

lowest terms. If �

� is a root of the polynomial�() = � +

����� + ����� +⋯�� + ��, � ≠ 0with integral

coefficients. Then, b is factor of constant term �� and c is a

factor of the leading coefficient�.

3. An nth degree polynomial can have at most n real roots.

4. Remainder Theorem: Let !() be any polynomial of degree

greater than or equal to one and a, be any real number. If

!()is divided by ( − �), then the remainder is equal to !(�).

5. Factor Theorem: Let !()be a polynomial greater than or

equal to 1 and a, be a real number such that !(�) = 0, then

( − �) is factor of !(). Conversely if ( − �)is factor of !(),

then !(�) = 0.

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Class X

Relationship between the zeroes and coefficients of a

polynomial

Let the zeroes of a quadratic polynomial be��#�$. Then

%&'(�)*+(*% = −�,-../�/-0,.1

�,-../�/-0,.12�#�

!+(�&�3(�)*+(*% =�(#%3�#33*+'

�(*��4�4*#3(��

Relationship between the zeroes and coefficients of a quadratic

polynomial

Let the zeroes of a quadratic polynomial be��#�$. Then

%&'(�)*+(*% = −�,-../�/-0,.1

�,-../�/-0,.12�#�

!+(�&�3(�)*+(*% =�(#%3�#33*+'

�(*��4�4*#3(��

Remark: If ��#�$ are the zeroes of the quadratic polynomial �().

Then the polynomial �()is given by

�() = 5[� + (� + $) + �$]

�() = 5[� + (%&'(�+((3%) + (!+(�&�3(�+((3%)]

Relationship between the zeroes and coefficients of a cubic

polynomial

Let �, $�#�8 be the zeroes of the cubic polynomial �() =

�3 + �2 + � + �, � ≠ 0 . Then by factor theorem − �, −

$, − 8are factor of �(). Also �(), being a cubic polynomial

cannot have more than three linear factors.

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%&'(�)*+(*% = � + $ + 8 = −�(*��4�4*#3(��

�(*��4�4*#3(���#�

%&'(�)*+(*% = �$ + 8� + $8 =�(*��4�4*#3(�

�(*��4�4*#3(��

!+(�&�3(�)*+(*% = �$8 = −�(#%3�#33*+'

�(*��4�4*#3(��

Division Algorithm for polynomials

Dividend =Quotient * Divisor + Remainder