Polynomial Proofs (1 Viewer)

[khorne]

Which proofs relating to polynomials would be beneficial to know? i.e which ones would be likely to pop up in an exam?

I can think of maybe proving if a polynomial nth degree is equal to another for more than n values, they are the same polynomial. And proving conjugate root theorem.

 

[Trebla]

I've hardly seen any exam questions requiring a proof of a polynomials theorem. The only one that really gets asked occasionally is to prove the multiple root theorem.

 

[Hermes1]

This one comes up quite a lot, I have seen it in a few trial papers:

Prove that

has no multiple roots for n is greater than or equal to 1.

 

[Hermes1]

I've hardly seen any exam questions requiring a proof of a polynomials theorem. The only one that really gets asked occasionally is to prove the multiple root theorem.

the multiple root theorem is that this one:

let
and then you diffentiate and show P(alpha) is also equal to zero.

 

[Timothy.Siu]

This one comes up quite a lot, I have seen it in a few trial papers:

Prove that

has no multiple roots for n is greater than or equal to 1.

taylor expansion of with terms!
and i remember doing that question too lol.

 

[Hermes1]

taylor expansion of with terms!
and i remember doing that question too lol.

i dont know what that taylor expansion thing is, but u definitely dont need that to solve that question

 

[khorne]

Yeah that's straight forward proof there. Off the top of my head, you differentiate, then the condition of them being multiple roots is p(0) = 0, but in the initial thing p(0) =/= 0? SOmething like that?

 

[Drongoski]

This one comes up quite a lot, I have seen it in a few trial papers:

Prove that

has no multiple roots for n is greater than or equal to 1.

If is a multiple root of P(x) = 0 then:







But





a contradiction



Note: above requires n >= 1



Edit

In what sense a contradiction? - the assumption there is a multiple root .

 

[jeel]

This one comes up quite a lot, I have seen it in a few trial papers:

Prove that

has no multiple roots for n is greater than or equal to 1.

how do you do it?

 

[Drongoski]

Which proofs relating to polynomials would be beneficial to know? i.e which ones would be likely to pop up in an exam?

I can think of maybe proving if a polynomial nth degree is equal to another for more than n values, they are the same polynomial. And proving conjugate root theorem.

Why don't you put up proof for this to make sure it's set out right.

 

[Hermes1]

If is a multiple root of P(x) = 0 then:







But





a contradiction



Note: above requires n >= 1

nice work. i had actually forgotten how to do this question, but luckily u posted

 

[khorne]

P(x) = a0 + a1x +...+ anx^n
Let z be a root, such that a0 + a1z +...+anz^n = 0
Now, if z is a root, that conjugate of both sides noting 0* = 0
(P(x)*) = 0* = 0
a0* + a1*z* + ...+an*z^n*
Conjugate of constants is constant, and conjugate of power is the same as power of conjugate
therefore a0 + a1z* +...+anz*^n = 0, i.e z* is a root.

 

[Drongoski]

P(x) = a0 + a1x +...+ anx^n
Let z be a root, such that a0 + a1z +...+anz^n = 0
Now, if z is a root, that conjugate of both sides noting 0* = 0
(P(x)*) = 0* = 0
a0* + a1*z* + ...+an*z^n*
Conjugate of constants is constant, and conjugate of power is the same as power of conjugate
therefore a0 + a1z* +...+anz*^n = 0, i.e z* is a root.

Good. You have the key ideas correct. Would have been helpful if you'd given your proof the LaTeX treatment.

 

[funnytomato]

Good. You have the key ideas correct. Would have been helpful if you'd given your proof the LaTeX treatment.

that's gonna take ages...