Polynomials in several variables
Let F in k[x1,x2,...,xn] be a polynomial in n variables. Then if F(a1,a2, ..., an) =0, I need to show that there exists polynomials Gi such that
F = sum (i=1 to n) [xi - ai] Gi
Could anyone help? :-)
F = sum (i=1 to n) [xi - ai] Gi
Could anyone help? :-)
Original post by Ishika
Let F in k[x1,x2,...,xn] be a polynomial in n variables. Then if F(a1,a2, ..., an) =0, I need to show that there exists polynomials Gi such that
F = sum (i=1 to n) [xi - ai] Gi
Could anyone help? :-)
F = sum (i=1 to n) [xi - ai] Gi
Could anyone help? :-)
Consider F as a polynomial in x1, and apply the remainder theorem.
What do you know about the remainder term?
Can you see how to continue?
Original post by Ishika
if F is a polynomial in x1 and F(a1)=0 then x1-a1 must divide F...
True, but not relevant, since you don't know that F(a1)= 0, as much as F(a1) makes any sense in this context.
Focus on the remainder theorem, not the factorisation theorem, also what is, or more accurately isn't in the remainder.
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