2. Table Of Contents -
1. History Of Notation
• Types Of Polynomials
• Zeroes Of Polynomial
• Graphs Of Polynomial Function
• Algebraic Identities
• Arithmetic Of Polynomials
• Think Tanker ? ?
~ What Is a Polynomial ??
In mathematics, a polynomial is an expression
consisting of variables and coefficients, that
involves only the operations of addition,
subtraction, multiplication, and non-negative
Example : x2 − 3x + 6, which is a quadratic
4. History Of Notation
• He popularized the use of letters from the
beginning of the alphabet to denote constants and
letters from the end of the alphabet to denote
• As can be seen above, in the general formula for a
polynomial in one variable, where the a's denote
constants and x denotes a variable.
• Descartes introduced the use of superscripts to
denote exponents as well.
6. Types Of Polynomials...
A monomial is a
just one term.
3x,4xy is a
In algebra, A
binomial is a
is the sum of two
2x+5 is a
trinomial is a
consisting of three
For Example :
3x+5y+7z is a
• Polynomials appear in a wide variety of areas of
mathematics and science.
~ For example, they are used to form “Polynomial”
equations, which encode a wide range of problems, from
elementary word problems to complicated problems in the
• They are used to define “Polynomial Functions”, which
appear in settings ranging from basic chemistry and
physics to economics and social science.
• They are used in calculus and numerical analysis to
approximate other functions.
9. Zeroes Of Polynomial
• Consider the polynomial p(x) = 5x3 – 2x3+ 3x – 2.
If we replace x by 1 everywhere in p(x), we get
p(1) = 5 × (1)3 – 2 × (1)2 + 3 × (1) – 2
= 5 – 2 + 3 –2
So, we say that the value of p(x) at x = 1 is 4.
p(0) = 5(0)3 – 2(0)2 + 3(0) –2
• The degree of a polynomial is the highest
degree of its terms, when the polynomial is
expressed in canonical form (i.e., as a linear
combination of monomials). The degree of a
term is the sum of the exponents of the
variables that appear in it.
13. Look at each term,
whoever has the most letters wins!
x2 – 4x4 + x6
This is a 8th degree polynomial:
xy4 + x4y4 + 12
This guy has 6 letters…
The degree is 6.
This guy has 8 letters…
The degree is 8
Here’s how you find the degree
of a polynomial :
14. The graph of the zero polynomial
f(x) = 0 is the x-axis.
Graphs Of Polynomial
15. The graph of the polynomial of degree 2
Graphs Of Polynomial
19. Addition Of Polynomials…..
• Polynomials can be added using the associative law of
addition (grouping all their terms together into a single
sum), possibly followed by reordering, and combining of
like terms. For example, if
Method 1: Line up like terms. Then add the coefficients.
P = 3x + 7
Q = 2x + 3
P + Q = 5x + 10
20. Addition Of Polynomials…..
Method 2 :
Group like terms. Then add the coefficients.
4x2 + 6x + 7 + 2x2 – 9x + 1 = (4x2 + 2x2)+(6x – 9x)+ (7+1)
= 6x2 – 3x + 8
» The sum of two polynomials is also a polynomial.
21. Subtraction Of Polynomials
• Earlier you learned that subtraction means to add
the opposite. So when you subtract a polynomial,
change the signs of each of the terms to its opposite.
Then add the coefficients.
Line up like terms. Change the signs of the second
polynomial, then add. For Example:
4x - 7 4x - 7
-(2x + 3) -2x – 3
2x - 10
22. Subtraction Of Polynomials
Simplify: (5x2 – 3x) – (-8x2 + 11)
Write the opposite of each term :
5x2 – 3x + 8x2 – 11
Group like terms :
(5x2 + 8x2) + (3x + 0) + (-11 + 0) = 13x2 + 3x – 11
»The difference of two polynomials is also a polynomial
24. Long- Division Method
• In arithmetic, long division is a standard division
algorithm suitable for dividing multi-digit numbers
that is simple enough to perform by hand.
• It breaks down a division problem into a series of
• As in all division problems, one number, called the
dividend, is divided by another, called the divisor,
producing a result called the quotient.
Dividend = (Divisor × Quotient) + Remainder
• Factor Theorem : If p(x) is a polynomial of
degree n > 1 and a is any real number,
(i) x – a is a factor of p(x), if p(a) = 0,
(ii) p(a) = 0, if x – a is a factor of p(x).
28. Q.1 What is the simplified form of :
29. Q.2 What is the value of x when x+3=10 ??