This presentation discusses polynomials and their key properties. It defines what polynomials are, including their terms, degrees, and different types (constant, linear, quadratic, cubic). It explains the relationship between the zeros of a polynomial and its coefficients. Specifically, it states that the sum of the zeros is equal to the negative of the coefficient of x^1, and the product of the zeros is equal to the constant term. It also discusses how to find the zeros of a polynomial by setting it equal to 0 and factoring. Examples are provided to illustrate various polynomial concepts and properties.

1. A
presentation
on
Presented By;-
NAME – Ashish Pradhan , Durgesh Kumar
CLASS- X – ‘A’
ROLL NO-27 , 26

2. INTRODUCTION
GEOMETRICAL MEANING
OF ZEROES OF THE
POLYNOMIAL
RELATION BETWEEN
ZEROES AND COEFFICIENTS
OF A POLYNOMIAL
DIVISION ALGORITHM
FOR POLYNOMIAL

3.  Polynomials are algebraic expressions that include real numbers and
variables. The power of the variables should always be a whole
number. Division and square roots cannot be involved in the
variables. The variables can only include addition, subtraction and
multiplication.
Polynomials contain more than one term. Polynomials are the sums
of monomials.
A monomial has one term: 5y or -8x2 or 3.
A binomial has two terms: -3x2 2, or 9y - 2y2
A trinomial has 3 terms: -3x2 2 3x, or 9y - 2y2 y
 The degree of the term is the exponent of the variable: 3x2 has a
degree of 2.
When the variable does not have an exponent - always understand
that there's a '1' e.g., 1x
Example:
x2 - 7x - 6
(Each part is a term and x2 is referred to as the leading term)

4. WHAT IS A POLYNOMIAL

5. Let “x” be a variable and “n” be a positive
integer and as, a1,a2,….an be constants
(real nos.)
Then, f(x) = anxn+ an-1xn-1+….+a1x+xo
 anxn,an-1xn-1,….a1x and ao are known as the
terms of the polynomial.
 an,an-1,an-2,….a1 and ao are their
coefficients.
For example:
• p(x) = 3x – 2 is a polynomial in variable x.
• q(x) = 3y2 – 2y + 4 is a polynomial in variable y.
• f(u) = 1/2u3 – 3u2 + 2u – 4 is a polynomial in variable u.
NOTE: 2x2 – 3√x + 5, 1/x2 – 2x +5 , 2x3 – 3/x +4 are not polynomials.

7. The degree is the term with the greatest exponent
Recall that for y2, y is the base and 2 is the exponent
For example:
 p(x) = 10x4 + ½ is a polynomial in the variable
x of degree 4.
 p(x) = 8x3 + 7 is a polynomial in the variable x
of degree 3.
 p(x) = 5x3 – 3x2 + x – 1/√2 is a polynomial in
the variable x of degree 3.
 p(x) = 8u5 + u2 – 3/4 is a polynomial in the
variable x of degree 5.

8. More information of degree

9. A real number α is a zero
of a polynomial f(x), if f(α)
= 0.
e.g. f(x) = x³ - 6x² +11x -6
f(2) = 2³ -6 X 2² +11 X 2
– 6
= 0 .
Hence 2 is a zero of f(x).
The number of zeroes of
the polynomial is the
degree of the polynomial.
Therefore a quadratic
polynomial has 2 zeroes
and cubic 3 zeroes.

10. For example:
f(x) = 7, g(x) = -3/2, h(x) = 2
are constant polynomials.
The degree of constant polynomials is ZERO.
For example:
 p(x) = 4x – 3, p(y) = 3y
are linear polynomials.
Any linear polynomial is in
the form ax + b, where a, b
are real nos. and a ≠ 0.
It may be a monomial or a
binomial. F(x) = 2x – 3 is binomial
whereas g (x) = 7x is monomial.

11.  A polynomial of degree two is
called a quadratic polynomial.
 f(x) = √3x2 – 4/3x + ½, q(w) =
2/3w2 + 4 are quadratic
polynomials with real
coefficients.
Any quadratic polynomial is
always in the form:-
ax2 + bx +c where a,b,c are real
nos. and a ≠ 0. • A polynomial of degree
three is called a cubic
polynomial.
• f(x) = 5x3 – 2x2 + 3x -1/5 is a
cubic polynomial in variable
x.
• Any cubic polynomial is
always in the form f(x = ax3
+ bx2 +cx + d where a,b,c,d
are real nos.

12. If p(x) is a polynomial and “y”
is any real no. then real no.
obtained by replacing “x” by
“y”in p(x) is called the value
of p(x) at x = y and is
denoted by “p(y)”.
A real no. x is a zero of the
polynomial f(x),is f(x) = 0
Finding a zero of the polynomial
means solving polynomial
equation f(x) = 0.
For example:-
Value of p(x) at x = 1
p(x) = 2x2 – 3x – 2
 p(1) = 2(1)2 – 3 x 1 – 2
= 2 – 3 – 2
= -3
For example:-
Zero of the polynomial
f(x) = x2 + 7x +12
 f(x) = 0
x2 + 7x + 12 = 0
(x + 4) (x + 3) = 0
x + 4 = 0 or, x + 3 = 0
x = -4 , -3
ZERO OF A POLYNOMIAL

13. ☻ A + B = - Coefficient of x
Coefficient of x2
= - b
a
☻ AB = Constant term
Coefficient of x2
= c
a
Note:- “A” and
“B” are the
zeroes.

14. Number of real zeroes of a
polynomial is less than or equal to
degree of the polynomial.
An nth degree polynomial can have at most “n”
real zeroes.

15. GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x + 2
LINEAR FUNCTION
DEGREE =1
MAX. ZEROES = 1

16. GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
QUADRATIC
FUNCTION
DEGREE = 2
MAX. ZEROES = 2

17. Relationship between the zeroes and coefficients of a cubic
polynomial
• Let α, β and γ be the zeroes of the polynomial ax³ + bx² + cx • Then, sum of zeroes(α+β+γ) = -b = -(coefficient of x²)
a coefficient of x³
αβ + βγ + αγ = c = coefficient of x
a coefficient of x³
Product of zeroes (αβγ) = -d = -(constant term)
a coefficient of x³

18. GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
CUBIC FUNCTION
DEGREE = 3
MAX. ZEROES = 3

20. If p(x) and g(x) are any two polynomials with
g(x) ≠ 0,then we can always find polynomials
q(x), and r(x) such that :
P(x) = q(x) g(x) + r(x),
Where r(x) = 0 or degree r(x) < degree g(x)

21. QUESTIONS BASED ON
POLYNOMIALS
I) Find the zeroes of the polynomial x² + 7x + 12and verify the relation between the
zeroes and its coefficients.
f(x) = x² + 7x + 12
= x² + 4x + 3x + 12
=x(x +4) + 3(x + 4)
=(x + 4)(x + 3)
Therefore,zeroes of f(x) =x + 4 = 0, x +3 = 0 [ f(x) = 0]
x = -4, x = -3
Hence zeroes of f(x) are α = -4 and β = -3.

23. 2) Find a quadratic polynomial whose zeroes are 4, 1.
sum of zeroes,α + β = 4 +1 = 5 = -b/a
product of zeroes, αβ = 4 x 1 = 4 = c/a
therefore, a = 1, b = -4, c =1
as, polynomial = ax² + bx +c
= 1(x)² + { -4(x)} + 1
= x² - 4x + 1

24. THE END