The document provides a comprehensive overview of polynomials, defining them as algebraic expressions with non-negative integral powers. It explains the concepts of polynomial degree, classification by number of terms, and various theorems such as the Remainder Theorem and the Factor Theorem. Additionally, it includes examples, algebraic identities for factorization, and key points regarding polynomial roots and zeros.

1. Polynomials
By : Saloni Garg

2. Introduction
An algebraic expression in which variables involved have only
non-negative integral powers is called a polynomial.
E.g.- (a) 2x3
–4x2
+6x–3 is a polynomial in one variable x.
(b) 8p7+4p2+11p3-9p is a polynomial in one variable p.
(c) 4+7x4/5
+9x5 is an expression but not a polynomial
since it contains a term x4/5
, where 4/5
is not
a non-negative integer.

3. Degree of a Polynomial in one
variable.
• What is degree of the following binomial?
3
5 2

x
The answer is 2. 5x2
+ 3 is a polynomial in x of degree 2.
In case of a polynomial in one variable, the highest power of the
variable is called the degree of polynomial.

4. Degree of a Polynomial in two
variables.
• What is degree of the following polynomial?
4
9
3
7
5 3
3
2



 y
xy
x
y
x
In case of polynomials on more than one variable, the sum of
powers of the variables in each term is taken up and the highest
sum so obtained is called the degree of polynomial.
• The answer is five because if we add 2
and 3
, the answer is five
which is the highest power in the whole polynomial.
E.g.- is a polynomial in x
and y of degree 7.
9
2
8
5
3 2
4
3



 y
xy
x
y
x

5. Polynomials in one variable
• A polynomial is a monomial or a sum of monomials.
• Each monomial in a polynomial is a term of the
polynomial.
The number factor of a term is called the coefficient.
The coefficient of the first term in a polynomial is the
lead coefficient.
• A polynomial with two terms is called a binomial.
• A polynomial with three term is called a trinomial.

6. Polynomials in one variable
The degree of a polynomial in one variable is the largest
exponent of that variable.
1
4 
x
A constant has no variable. It is a 0 degree polynomial.
2
This is a 1st degree polynomial. 1st degree polynomials
are linear.
14
2
5 2

 x
x This is a 2nd degree polynomial. 2nd
degree polynomials are quadratic.
18
3 3

x This is a 3rd degree polynomial. 3rd degree
polynomials are cubic.

7. Examples
Text
Text
Txt
Text
Text
Polynomials Degree Classify by
degree
Classify by no.
of terms.
5 0 Constant Monomial
2x - 4 1 Linear Binomial
3x2
+ x 2 Quadratic Binomial
x3
- 4x2
+ 1 3 Cubic Trinomial

8. Standard Form
Phase 1 Phase 2
To rewrite a
polynomial in
standard form,
rearrange the
terms of the
polynomial
starting with the
largest degree
term and ending
with the lowest
degree term.
The leading coefficient,
the coefficient of the
first term in a
polynomial written in
standard form, should
be positive.
How to convert a polynomial into standard form?

9. Remainder Theorem
TEXT TEXT TEXT TEXT
Let f(x) be a polynomial of degree n > 1 and let a be any real number.
When f(x) is divided by (x-a) , then the remainder is f(a).
PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder
is r(x).
Then, degree r(x) < degree (x-a)
degree r(x) < 1 [ therefore, degree (x-a)=1]
degree r(x) = 0
r(x) is constant, equal to r (say)
Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.
Therefore, f(x) = (x-a)*g(x) + r (i)
Putting x=a in (i), we get r = f(a)
Thus, when f(x) is divided by (x-a), then the remainder is f(a).

10. Questions on Remainder Theorem
Q.) Find the remainder when the polynomial
f(x) = x4
+ 2x3
– 3x2
+ x – 1 is divided by (x-2).
A.) x-2 = 0 x=2
By remainder theorem, we know that when f(x) is divided by (x-2),
the remainder is x(2).
Now, f(2) = (24
+ 2*23
– 3*22
+ 2-1)
= (16 + 16 – 12 + 2 – 1) = 21.
Hence, the required remainder is 21.

11. Factor Theorem
Let f(x) be a polynomial of degree n > 1 and let a be
any real number.
(i) If f(a) = 0 then (x-a) is a factor of f(x).
PROOF let f(a) = 0
On dividing f(x) by 9x-a), let g(x) be the quotient. Also,
by remainder theorem, when f(x) is divided by (x-a),
then the remainder is f(a).
therefore f(x) = (x-a)*g(x) + f(a)
f(x) = (x-a)*g(x) [therefore f(a)=0(given]
(x-a) is a factor of f(x).

12. Algebraic Identities
Some common identities used to factorize polynomials
(x+a)(x+b)=x2+(a+b)x+ab
(a+b)2
=a2
+b2
+2ab (a-b)2
=a2
+b2
-2ab a2
-b2
=(a+b)(a-b)

13. Algebraic Identities
Advanced identities used to factorize polynomials
(x+y+z)2
=x2
+y2
+z2
+2xy+2yz+2zx
(x-y)3
=x3
-y3
-
3xy(x-y)
(x+y)3
=x3
+y3
+
3xy(x+y)
x3
+y3
=(x+y) *
(x2
+y2
-xy) x3
-y3
=(x+y) *
(x2
+y2
+xy)

14. Q/A on Polynomials
Q.1) Show that (x-3) is a factor of polynomial
f(x)=x3+x2-17x+15.
A.1) By factor theorem, (x-3) will be a factor of f(x) if f(3)=0.
Now, f(x)=x3+x2-17x+15
f(3)=(33+32-17*3+15)=(27+9-51+15)=0
(x-3) is a factor of f(x).
Hence, (x-3) is a factor of the given polynomial f(x).

15. Q/A on Polynomials
Q.1) Factorize:
(i) 9x2 – 16y2 (ii)x3-x
A.1)(i) (9x2 – 16y2) = (3x)2 – (4y)2
= (3x + 4y)(3x – 4y)
therefore, (9x2-16y2) = (3x + 4y)(3x – 4y)
(ii) (x3-x) = x(x2-1)
= x(x+1)(x-1)
therefore, (x3-x) = x(x + 1)(x-1)

16. Points to Remember
• A real number ‘a’ is a zero of a polynomial p(x) if p(a)=0. In
this case, a is also called a root of the equation p(x)=0.
• Every linear polynomial in one variable has a unique zero, a
non-zero constant polynomial has no zero, and every real
number is a zero of the zero polynomial.