Division of multivariable polynomials.
- Thread starter askan
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Dr.Peterson
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It will not divide evenly, as you can see by letting x=3 and y=-2; if (2x+3y) were a factor, you would get 0. So you can expect a remainder.
But also, the second coefficient in your quotient should be -9/2 in order to eliminate -9x^3y. You are, in effect, treating x as the variable and y as a mere parameter, so you need to have a lower degree in x at each step.
Um so yeah is this a method for telling if we can expect a remainder from a division??. If it is then can you tell me more about it.
ok got it. fractional quotients are to be used here.
umm can you explain this a bit more because I didn't understand this part at all.
Dr.Peterson
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Yes. Have you learned the factor theorem? We're just extending it to multiple variables.
In a sense it should be obvious. If you can write f(x,y) = (2x+3y)(some polynomial), then f(3,-2) = (0)(something) = 0.
You wrote the dividend in order of decreasing degree in x. So you were doing what you would do for 6x^4 - x^2 +x + 4, ignoring the y. But, of course, you can't totally ignore it! I see what you did as dividing (6)x^4 - (y)x^2 +(y^2)x + (4y^4), where the coefficients are 6, -y, y^2, and 4y^4. We're treating y as if it were a constant. That's called a parameter.
And at each step of a division, you want to eliminate a term entirely, so you have to subtract the entire (-9y)x^3 in order to make progress. That's how long division works.