i) The document discusses the relation between the roots and coefficients of polynomials. The sum and product of roots are related to the coefficients in the polynomial. ii) It also discusses the nature of roots of quadratic equations. The roots are real and equal if the discriminant is 0, real and distinct if the discriminant is positive, and imaginary if the discriminant is negative. iii) Examples are given to illustrate cases when the roots are real/equal, real/distinct, and imaginary.

1. MATHEMATICAS
i)Relation between roots and the coefficients.
ii)Nature of the roots of quadratic equations.
Presentation by
D.Rohini
I B.Sc. Mathematics(R)

2. RELATION BETWEEN THE ROOTS AND COEFFICIENTS
Let 𝜶 𝟏, 𝜶 𝟐, … 𝜶 𝒏 be the roots of the equation
𝒂 𝟎 𝒙 𝒏 + 𝒂 𝟏 𝒙 𝒏−𝟏 + 𝒂 𝟐 𝒙 𝒏−𝟐 + ⋯ 𝒂 𝒏
𝑺 𝟏 = 𝒊=𝟏
𝒏
𝜶𝒊 = (−𝟏)
𝒂 𝟏
𝒂 𝟎
𝑺 𝟐 = 𝒊,𝒋
𝒏
𝜶𝒊 𝜶𝒋 = (−𝟏) 𝟐 𝒂 𝟐
𝒂 𝟎
𝑺 𝒏 = 𝜶 𝟏, 𝜶 𝟐, … , 𝜶 𝒏 = (−𝟏) 𝒏 𝒂 𝒏
𝒂 𝟎
where 𝑺 𝒏denotes the sum of the product of n roots at a time.

3. NATURE OF THE ROOTS OF A QUADRATIC EQUATION
The quadratic equation : 𝒂𝒙 𝟐
+ 𝒃𝒙 + 𝒄 = 𝟎
The roots are given by
𝒙 =
−𝒃± 𝒃 𝟐−𝟒𝒂𝒄
𝟐𝒂

4. NATUREOFTHEROOTSOFAQUADRATIC
EQUATION
Let ∆ = 𝒃 𝟐
− 𝟒𝒂𝒄
Case i)if ∆ = 𝑏2 − 4𝑎𝑐 = 0,then the roots are real and
equal.
Case ii)if ∆ = 𝑏2 − 4𝑎𝑐 > 0,then the roots are real and
distinct.
Case iii)if ∆ = 𝑏2 − 4𝑎𝑐 <0,then the roots are
imaginary.

5. EXAMPLES
Case I : 𝒙 𝟐 + 𝟒𝒙 + 𝟒 = 𝟎
here ∆ = 𝑏2 − 4𝑎𝑐 = 0,therefore the roots are real and
equal.
Case II : 𝒙 𝟐 + 𝟓𝒙 + 𝟔 = 𝟎
here ∆ = 𝑏2
− 4𝑎𝑐 > 0,therefore the roots are real and
distinct.
Case III : 𝒙 𝟐 + 𝒙 + 𝟏 = 𝟎
here ∆ = 𝑏2 − 4𝑎𝑐 <0,therefore the roots are
imaginary.