Question

Condition for a cubic equation to have three real and distinct roots using calculus

Answer 1

a cubic polynomial has at most three zeros

Answer 2

Answer:

One way could be as follows.

Take f(x) = ax³ + bx² + cx + d.

There are 3 distinct real root

<=> the graph of f(x) has a local min on one side of the x-axis and a local max on the other side

<=> the quadratic f'(x) has real roots x₁ and x₂, and f(x₁) and f(x₂) have opposite signs

So we go

f(x) = ax³ + bx² + cx + d

=> f'(x) = 3ax² + 2bx + c

This must have real roots, so condition 1 is 4b² - 12 ac > 0, or equivalently,

Condition 1: b² > 3ac

Given that this is satisfied, the roots are

x₁ = ( -2b - √(4b² - 12ac) ) / 6a   =  ( -b - √(b² - 3ac) ) / 3a

and

x₂ = ( -2b + √(4b² - 12ac) ) / 6a  = ( -b + √(b² - 3ac) ) / 3a

Then we have

Condition 2: f(x₁) and f(x₂) have different signs

This could be reworded in a purely algebraic form as

Condition 2: f(x₁) f(x₂) < 0