roots of cubics

wildrabbitt · 2019-09-25, 17:15

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

R.D. Silverman · 2019-09-25, 17:26

Quote:

Originally Posted by wildrabbitt

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

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.

henryzz · 2019-09-25, 17:28

Quote:

Originally Posted by wildrabbitt

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

It is possible to have repeated roots of cubic equations.

fivemack · 2019-09-25, 17:38

Quote:

Originally Posted by wildrabbitt

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

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 )

R.D. Silverman · 2019-09-25, 18:25

Quote:

Originally Posted by fivemack

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 )

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!

VBCurtis · 2019-09-25, 18:54

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.

R.D. Silverman · 2019-09-25, 19:16

Quote:

Originally Posted by VBCurtis

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.

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

R.D. Silverman · 2019-09-25, 19:18

Quote:

Originally Posted by R.D. Silverman

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

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

fivemack · 2019-09-25, 19:20

Quote:

Originally Posted by R.D. Silverman

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

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

R.D. Silverman · 2019-09-25, 19:24

Quote:

Originally Posted by fivemack

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

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.

VBCurtis · 2019-09-25, 20:01

Quote:

Originally Posted by R.D. Silverman

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

Two bad assumptions, even around here.

2019-09-25, 17:15	#1
wildrabbitt Jul 2014 3×149 Posts	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

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2019-09-25, 18:54	#6
VBCurtis "Curtis" Feb 2005 Riverside, CA 2·2,339 Posts	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.