Question

Polynomial question halp!!!!!!!!!!!- show that, if b^2<3ac, the polynomial ax^3 +bx^2 +cx +d cannot have a multiple root

Answer 1

What does not having multiple root means ? Is it not having more than 1 real root ? or not having any equal roots ?

Answer 2

i think no equal roots

Answer 3

because im not doing complex roots atm

Answer 4

if you let
f(x) = ax^3 +bx^2 +cx +d
then
f'(x) = 3ax^2 + 2bx + c

the discriminant of f'(x) is given to be <0
right so far ?

Answer 5

why do we need it to be less than 0?

Answer 6

need it to be ?
we are given it is less than 0, and we then have to prove multiple roots dont exist

Answer 7

but here it is b^2 -3ac<0

Answer 8

discriminant = 4b^2 - 12ac = 4(b^2 - 3ac) i.e. <0
so we know discriminant of f'(x) is <0.

we are clear so far ?

Answer 9

ohhh yup. all good

Answer 10

When discrim. is <0 , it means that quadratic eqn has no real roots

since f'(x) has no real roots or f'(x) is never equal to 0, we can conclude the cubic eqn f(x) has no maxima or minima right ?

Answer 11

yup

Answer 12

A cubic is with no maxima or minima is always either increasing or decreasing. Right ?
(We don't need to find in this case whether its increasing or decreasing)

Answer 13

yeh thats right

Answer 14

So it means the graph must be going from -inf to +inf and must have crossed x-axis only once somewhere.
Hence it has only 1 real root.
I guess not having multiple roots means not more than 1 real root.

Answer 15

ahhh yeh i think it was a typo

Answer 16

thank you so much, you are so helpful :D

Answer 17

glad to help.