Roots of a polynomial function
If 
and 
are roots of quadratic equation
Then (x – 
) (x – β) = 0
x2 – βx – 
x + 
β = 0
x2 – (β +
) x + 
β = 0
Given a quadratic equation as
ax2 + bx + c = 0, where a, b, c, are constant
Summary
A quadratic equation is given by
x2 – (sum of factors) x + products of factors = 0
Example
1. Given 
and β as the roots for 4x2 + 8x + 1 = 0 form an equation whose roots are 
2 β and β2
Solution
Sum of roots 
2 β + β2 
= 
β (
+ β)
Products of root are (
2 β) (β2
)
=
3 β3
= (
β) 3
(
2 β) (β2
) = (
β) 3
The given equation can be written as
The required equation is
=0
2. The equation 3x2 – 5 + 1 = 0 has roots 
and β
a) Find values of 
b) Find the values of 
Solution
+ β = 
and 
β = 
Roots of cubic equations
If
, β, 
are roots of a cubic equation then
(x –
)(x – β)(x – γ) = 0
(x2 – 
x – βx + 
β) (x – γ) = 0
x3 – γx2 – 
x2 + 
γx – βx2 + βγx + 
βx – 
βγ = 0
x3 – (
+ β + γ) x2 + (
γ+ βγ + 
β) x – 
βγ = 0
x3 – (
+ β + γ) x2 + (
γ + βγ + 
β) x – 
βγ = 0
Given cubic equation can be written as
ax3 + bx2 + cx + d = 0
Equating coefficients of x2, x and the constant terms
i) 
+ β + γ = 
-; sum of roots
ii) 
γ + βγ + 
β = 
; sum of products of roots
iii) 
γβ = 
; products of roots
Examples
1. The equation 3x3 + 6x2 – 4x + 7 = 0 has roots
, β, γ. Find the equations with roots
a)
Solution
From
x3 – (sum of factors) + (sum of products of factors) – products = 0
X3 – (sum of factors) x2 + (sum of products of products of factors) x – products = 0
From the equation 3x3 + 6x2 – 4x + 7 = 0
2. If the roots of the equation 4x3 + 7x2 – 5x – 1 = 0 are 
, β and γ find the equation whose roots are
a)
+ 1, β + 1, γ + 1 b) 
2, β2, γ2
Solution
4x3 + 7x2 – 5x – 1 = 0
