July 31, 2018 Tuesday
Bedtime Story
Derivation of the Roots of Depressed Cubic Equation
Today we shall study the so called
Cardano’s method (I say so called because it was Scipione del Ferro who was the
real brain behind the solution) for arriving at the roots of the depressed
cubic equation
t3 + pt + q = 0
The very first step of del Ferro in the
solution was to introduce two variables u and v such that
u +
v = t
Now let us see what happens when we
introduce the two variables u and v in the depressed cubic equation.
It gives us
u3 + v3 + (3uv + p)(u + v) + q = 0
Here Cardano introduced a second condition on
the variables u and v so that 3uv + p = 0
Imposing this condition on the above
equation will give us
u3 + v3 = - q
From 3uv + p = 0 we will also derive
u3v3 = - p3/27
There two above equations are the sum and
product of u3 and v3 and hence once can derive a
quadratic equation from them.
The quadratic equation to which u3
and v3 are the solutions or roots are:
z2 + pz – p3/27
= 0
Solving the quadratic equation gives
u3 = -q/2 + √(q2/4 + p3/27)
v3 = -q/2 - √(q2/4 + p3/27)
Since t = u + v, then
t = cubic root of (-q/2 + √(q2/4 + p3/27) + cubic
root of (-q/2 - √(q2/4 + p3/27)
This is the Cardano’s solution for the
depressed cubic equation and you can see even it required a high level of
ingenuity and three brilliant mathematical minds to arrive at the solution.
Now lastly, we shall look into the quartic
function which has the form
f(x) = ax4 + bx3 + cx2 + dx + e where a
≠ 0
A quartic function or a quartic polynomial
can be converted to a quartic equation by equating it to zero.
Thus a quartic equation looks like
ax4 + bx3 + cx2 + dx + e = 0 where a ≠
0
The derivative of a quartic function is a
cubic function.
The solution of the quartic equation was
published in the same book (Ars Magna) of Gerolamo Cardano in 1545 but its
solution to Cardano was aided by his bright servant Lodovico de Ferrari.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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