For a function ax^n+bx^(n-1)...
What is the proof that that function is the same as

a(x-r1)(x-r2).... where r are the roots of the function?

That is: how can we be sure that if a f(x)'s coefficient for x^n (where n is the highest) is the same as g(x)'s for x^n AND they share the same roots that they are the same function?

Question

For a function ax^n+bx^(n-1)...
What is the proof that that function is the same as

a(x-r1)(x-r2).... where r are the roots of the function?

That is: how can we be sure that if a f(x)'s coefficient for x^n (where n is the highest) is the same as g(x)'s for x^n AND they share the same roots that they are the same function?

experimentx · Answer

put r1,r2, .. in  ax^n+bx^(n-1).. <-- this should be zero (by the definition of root)

put r1,r2, ... in a(x-r1)(x-r2)... <-- this is obviously zero.

anonymous · Answer

I understand that far, so am I trying to be too rigourous in proving something obvious?

experimentx · Answer

$$ f(x) = a(x - r_1)(x-r_2) ...(x-r_n) = a( x^n + (-)(r_1+r_2 + .. +r_n)x^{n-1} + ... + r_1r_2 ..r_n $$.

experimentx · Answer

they are just different form of equation. A polynomial of nth degree has n roots means you can factor them into n factors. which is the product of x - root.

anonymous · Answer

I WAS overcomplicating. Thank you anyway