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 11:06, 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!