this one
#This one
undone osprey
blazing bridgeBOT
this one
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twin sable
Does it say « find the number of cubic polynomials that satisfy the given conditions »? I have a hard time reading
undone osprey
ya, find the no.
twin sable
Well there is only 1
You can prove it by taking two cubics P and Q that satisfy the conditions
Now you can consider P-Q and prove that P-Q is the 0 polynomial
(Think in terms of roots)
twin sable
That’s not true there is one and it’s unique
it’s Lagrange’s interpolation polynomials
undone osprey
twin sable
You can prove existence +unicity by doing some linear algebra by considering $P \to (P(1),P(2),P(3),P(4))$ you can prove that it’s a morphism from $\mathbb{R}_3[X]$ to $\mathbb{R}^{4}$ and prove that it’s actually an isomorphism
tacit umbraBOT
Rotor
undone osprey
i didnt understand this
twin sable
Now you can consider P-Q and prove that P-Q is the 0 polynomial
Ok no worries, hmmm how to explain this well we can actually construct the polynomial ourselves and prove unicity by using this
Maybe I misunderstood what needs to be proven
You need to find the number of cubics that satisfy the conditions right ?
undone osprey
ya
twin sable
Ok Well what we can do is consider the polynomials $L_1, L_2, L_3$ and $L_4$
tacit umbraBOT
Rotor
twin sable
Such that L_1(1)=1 and L_1(2)=L_1(3)=L_1(4)=0
And samething for the others you have $L_2(2)=1$ and $L_2(i)=0$ for $i \neq 2$
tacit umbraBOT
Rotor
undone osprey
i dont understand
twin sable
Ya know what I’ll just give the answer and explain why it works okay?
I think it’s better
undone osprey
okay
twin sable
If P=$-2\frac{(x-2)(x-3)(x-4)}{6}+4\frac{(x-1)(x-3)(x-4)}{2}-6\frac{(x-1)(x-2)(x-4)}{2} +8\frac{(x-1)(x-2)(x-3)}{6}$
tacit umbraBOT
Rotor
