Multiple Root Theorem (1 Viewer)

[Estel]

Does anyone have the proof to this or knows the direct link to a site with it... in language a Yr 11 non-4U student can understand? Don't have a 4U textbook on hand.

Thankyou.

 

[George W. Bush]

Is this the theorem that states that if A is a double root of f(x), then A is a root of f'(x)?

If so, it's quite easy to prove (off the top of my head (may be a better away))
f(x) = (x-A)^2 * Q(x)
f'(x) = (x-A)^2 * Q'(x) + 2(x-A)Q(x)
= (x-A)[(x-A)*Q'(x) + 2Q(x)]
i.e. x=A is a root of f'(x)

 

[:: ck ::]

yes that is how u prove it...
btw shud be Q'(x) in line 2 and 3

alternatively u can think of it graphically
if u have a dbl root... u kno the gradient is going to b zero

so for P(x) to have a dbl root at x = a
wotever value of x has the dbl root... P(a) = 0
and P'(a) = 0 [gradient is equal to zero]

likewise with roots of odd multiplicity which haf a pt of inflexion
( eg : triple root P(a) = P'(a) = P''(a) = 0 )

 

[George W. Bush]

ah yes, my mistake


edited

 

[Estel]

Thankyou Mr Bush... tho I read CM's version as I browsed the Mathematics section first ... essentially the same to answer your piece of speculation ("may be a better way")...

Ryan... pt of inflexion? stop using such jargon. I used your method to try and explain things to myself, but being a rather intuitive method, you can't really justify yourself in that manner.

 

[Xayma]

Originally posted by Estel
Ryan... pt of inflexion? stop using such jargon. I used your method to try and explain things to myself, but being a rather intuitive method, you can't really justify yourself in that manner.

Jargon you will have to learn it later, a point of inflexion is where concavity changes (found by the second derivative and used to tell if a statioary point is a maximum or minimum)

 

[Estel]

Yeh... my teacher told me last week lol... I was sick of all this nomenclature I didn't have a clue about...

It's irritatin to have your friends who go to tutor have lively conversations about the virtues of the product rule over the quotient rule, the need for radians in calculus and the definition of e when you don't have a clue. In good time...

 

[velox]

Originally posted by Estel
Yeh... my teacher told me last week lol... I was sick of all this nomenclature I didn't have a clue about...

It's irritatin to have your friends who go to tutor have lively conversations about the virtues of the product rule over the quotient rule, the need for radians in calculus and the definition of e when you don't have a clue. In good time...

its stupid to use the product rules instead of quotient, its so much easier to go wrong. damn ppl who think they're smart...

 

[:: ck ::]

Originally posted by Estel
Thankyou Mr Bush... tho I read CM's version as I browsed the Mathematics section first ... essentially the same to answer your piece of speculation ("may be a better way")...

Ryan... pt of inflexion? stop using such jargon. I used your method to try and explain things to myself, but being a rather intuitive method, you can't really justify yourself in that manner.

was just tryin to help u think of it in a graphical sense
rather than just knowing the theorem without knowing why it works