roots of cubics
 

Hi,


say a cubic equation intersects the y-axis at a point where it's derivative is 0 and also intersects the y-axis at one other point (so it touches the y-axis twice), wouldn't it have to have one root that isn't real?


Will

[QUOTE=wildrabbitt;526574]Hi,


say a cubic equation intersects the y-axis at a point where it's derivative is 0 and also intersects the y-axis at one other point (so it touches the y-axis twice), wouldn't it have to have one root that isn't real?


Will[/QUOTE]

If you mean: y = f(x) where f is a cubic polynomial:

IMPOSSIBLE. It is an elementary exercise to see why. There are two different reasons.

[QUOTE=wildrabbitt;526574]Hi,


say a cubic equation intersects the y-axis at a point where it's derivative is 0 and also intersects the y-axis at one other point (so it touches the y-axis twice), wouldn't it have to have one root that isn't real?


Will[/QUOTE]

It is possible to have repeated roots of cubic equations.

[QUOTE=wildrabbitt;526574]Hi,


say a cubic equation intersects the y-axis at a point where it's derivative is 0 and also intersects the y-axis at one other point (so it touches the y-axis twice), wouldn't it have to have one root that isn't real?


Will[/QUOTE]

No: if any polynomial has f(t)=0 and f'(t)=0 for the same t, that t is a multiple root of the polynomial (that is, the polynomial is divisible by (x-t)^2 )

[QUOTE=fivemack;526583]No: if any polynomial has f(t)=0 and f'(t)=0 for the same t, that t is a multiple root of the polynomial (that is, the polynomial is divisible by (x-t)^2 )[/QUOTE]


Sigh..... Bob runs screaming from the classroom...…...Has everyone forgotten basic
algebra????

Reread what the OP wrote!! He said that the curve itself hits the y-axis twice......Once
where its derivative is 0. i.e. he wants f(0) to have TWO DIFFERENT VALUES.
This is not a function!!! [y = cubic polynomial in x]

And, of course, a cubic can NEVER have a single imaginary root......Imaginary roots
come in pairs!

Bob-
It's pretty clear the OP meant x-axis, rather than y-axis. Your answer about imaginary (complex) roots always coming in pairs helps whether he typo'ed y-axis or not, though.

[QUOTE=VBCurtis;526593]Bob-
It's pretty clear the OP meant x-axis, rather than y-axis. Your answer about imaginary (complex) roots always coming in pairs helps whether he typo'ed y-axis or not, though.[/QUOTE]

I assume that people mean what they write. I assume that they proofread before posting.

[QUOTE=R.D. Silverman;526594]I assume that people mean what they write. I assume that they proofread before posting.[/QUOTE]

Note that he wrote 'y-axis' three different times.....

[QUOTE=R.D. Silverman;526595]Note that he wrote 'y-axis' three different times.....[/QUOTE]

This makes it even clearer that he meant 'the y=0 axis' (IE the X axis)

[QUOTE=fivemack;526596]This makes it even clearer that he meant 'the y=0 axis' (IE the X axis)[/QUOTE]

I disagree. The content of the post speaks for itself. I assume that people mean what they
write, especially when discussing a subject (such as mathematics) where it is possible
to always use precise language.

[QUOTE=R.D. Silverman;526594]I assume that people mean what they write. I assume that they proofread before posting.[/QUOTE]

Two bad assumptions, even around here.