Difference between revisions of "Multivariate factor theorem"
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− | ''' The Multivariable Factor Theorem '''states that If <math>f(x,y)</math> is a polynomial and there is a polynomial <math>h(x)</math> such that <math>f(x,h(x))=0</math> for | + | ''' The Multivariable Factor Theorem '''states that If <math>f(x,y)</math> is a polynomial and there is a polynomial <math>h(x)</math> such that <math>f(x,h(x))=0</math> for '''all''' <math>x,</math> then we can write |
<cmath> f(x,y) = (y-h(x))g(x,y),</cmath> for some polynomial <math>g(x,y).</math> | <cmath> f(x,y) = (y-h(x))g(x,y),</cmath> for some polynomial <math>g(x,y).</math> | ||
Latest revision as of 12:06, 15 March 2024
The Multivariable Factor Theorem states that If is a polynomial and there is a polynomial
such that
for all
then we can write
for some polynomial
Proof:
Assume that for all
. We'll treat
as a constant, so that
is constant with respect to
If we divide by
using polynomial long division, so that we have
Since we're treating as a constant,
is a monic, linear polynomial in
So, either
is the zero polynomial, in which case it has no terms with
or it has lower degree in
than
This means that
will itself be a polynomial in
Now, if we set in our equation, it becomes
It follows that
So for any
and so
is the zero polynomial!