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I'm just a 9th grader trying to self-study, so if the question sounds silly to you please excuse me.

$P(x) = x^3 + y^3$ is divided by something like $g(x) = x + 5;$ the degree of the polynomial is $3$. Doesn't the degree of the remainder always be less than that of the divisor. Can someone explain what is happening?

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    $\begingroup$ I believe the problem is simply that you're dealing with a polynomial in two variables, $x,y$ for $P$. That the remainder has a lesser degree than the divisor only really holds in the case of polynomials in one variable, I believe. $\endgroup$ – Eevee Trainer Feb 20 at 7:17
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    $\begingroup$ To rewrite my previous (now deleted) comment because I'm dumb and it's been a while since I've dealt with this. Basically I noticed you have $P(x)$, but $P(x,y)$ would be more appropriate if $y$ was a variable. If $y$ is not a variable, i.e. some constant, then your remainder (which I got to be $y^3 - 125$) would be of degree zero as intended in the one-variable case. Of course if we have multiple variables a lot of nice stuff we have simply flies out the window. Such is math. $\endgroup$ – Eevee Trainer Feb 20 at 7:23
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    $\begingroup$ The degree of that result viewed as a polynomial in x is less than 3. $\endgroup$ – William Elliot Feb 20 at 7:53
  • $\begingroup$ So, y is not a variable but just a constant that we do not know the value of.Since y is a constant its degree will be 0 and not 3. $\endgroup$ – Aditya Bharadwaj Feb 20 at 8:12
  • $\begingroup$ If you wish to learn how to extend the division algorithm to multivariate polynomails then search on "grobner basis". $\endgroup$ – Bill Dubuque Feb 20 at 21:52
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As clarified by the OP in the comments that $y$ is a constant. I would rewrite $y$ as $a$ so that it looks more constant-like for the sake of convenience.

Now $P(x)=x^3+a^3$ and $g(x)=x+5$. By Euclid's Division Lemma for Polynomials you do have $\text{deg}\ r(x)=0$ or $\text{deg }r(x)\lt\text{deg }g(x)$.

You may confirm this by long division method which gives you the following result: $$\dfrac{x^3+a^3}{x+5}=x^2+5x-25+\dfrac{125+a^3}{x+5}$$ or $r(x)=125+a^3$ and $q(x)=x^2+5x-25$ which clearly satisfies $\text{deg }g(x)=1\gt\text{deg }r(x)=0$. So there is no discrepancy.

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