Ch-2, Polynomials ppt

1. Welcom
e

2. TABLE OF
CONTENTS
01
04
03
02
To Polynomials
• On the basis of degree
• On the basis of term
Zeroes and coefficients of Quadratic
polynomials.
Number of zeroes a polynomial can have.

3. • A polynomial is defined as an expression which is
composed of variables, constants and exponents, that
are combined using the mathematical operations such
as addition, subtraction, multiplication and division.
• Examples of constants, variables and exponents are as
follows:
a) Constants Example: 1, 2, 3, etc.
b) Variables Example: g, h, x, y, etc.
c) Exponents Example: 5 in x5 etc.
• Example of a polynomial is 4x2+1x+3.

4. TYPES OF POLYNOMIALS
ON THE BASIS OF TERMS
ON THE BASIS OF DEGREE
LINEAR
POLYNOMIAL
QUADRATIC
POLYNOMIAL
CONSTANT
POLYNOMIAL
TRINOMIAL
BINOMIAL
MONOMIAL
CUBIC
POLYNOMIAL

5. • What is the Degree of a Polynomial ?
Ans: Thedegreeofapolynomialisthehighestsumoftheexponents/powersof
apolynomial’svariables.
Here,inthisexamplethehighest
powersofthegivenpolynomialsis
thewritteninthedegreecolumn.

6. ON THE BASIS OF
DEGREE
1. Constant polynomial
 A polynomial of degree zero is called a ‘Constant Polynomial’.
 Example – 7x
0
= 7 , 4x
0
= 4, etc.
 It is also called zero polynomial.
2. Linear polynomial
 A polynomial of degree 1 is called ‘Linear Polynomial’
 Example – 2x-3, 4x+9, etc.
 The most general form of linear polynomial is ax+b, where
a ≠ 0 and a & b are real.

7. 3. Quadratic Polynomial
 A polynomial with degree 2 is called a ‘Quadratic polynomial’.
 Example – 2x
2
+3x+6, x
2
- 2, etc.
 Generally, quadratic polynomial with x variable is written in the
form of ax2
+ bx + c, where a ≠0 and a, b & c are real.
4. Cubic Polynomial
 A polynomial of degree 3 is called ‘Cubic polynomial’.
 Example- 2-x3
, x3
– 1,etc.
 The most general form of cubic polynomials is ax3
+bx2
+cx+1
where a, b, c & d are real.

9. Let α , β and γ be the zeroes of the polynomial f(x)= ax²+bx+c.
By factor theorem (x- α)and (x- β) are the factors of f(x).
∴ f(x)=k(x- α)(x- β), where k is a constant.
Now putting all the values -
⇒ ax²+bx+c= k {x²-(α+ β )x+ αβ}
⇒ax²+bx+c= kx²-k(α+ β )x+ kαβ
Comparing the coefficients of x²,x and constant terms on both sides,
we get
a = k, b = -k(α+ β ) and c = kαβ
⇒ α+ β= -b/a and αβ=c/a
⇒ α+ β(Sum of the zeroes)= -(Coefficient of x)
Coefficient of x²
and αβ(Product of the zeroes)= Constant term
Coefficient of x²

10. • The graph of a linear polynomial intersects the
x-axis at a maximum of one point. Therefore, a
linear polynomial has a maximum of one zero.
• The graph of a quadratic polynomial intersects
the x-axis at a maximum of two points.
Therefore, a quadratic polynomial can have a
maximum of two zeroes.
• The graph of a cubic polynomial intersects the x-
axis at maximum of three points. A cubic
polynomial has a maximum of three zeroes.

11. THANKS
FOR
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