# Divide with remainder $\frac{x^2}{x^2 + x + 2}$

I am having a hard time long dividing: $$\frac{x^2}{x^2 + x + 2}.$$

Could someone please show a step by step way to divide this, as I can only get it down to : $1 + \frac{x^2}{x + 2}$.

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$1+{x^2\over x+2}$ is not correct. If x=0, then the original expression has the value 0, but your expression has the value 1, so they do not mean the same thing. – MJD Mar 14 '12 at 14:40
Try with $x=10$, i.e., $100/112$. It's not the same as $1+100/102$, is it? – lhf Mar 14 '12 at 14:58

You can do long division of polynomials exactly like one does long division of integers on paper.

Set it:

                       ___________________________
x^2+x+2      |   x^2


Now: look at the leading term of the divisor: $x^2$; the leading term of the dividend: $x^2$; how many times does the leading term of the divisor go into the leading term of the dividend? $x^2$ goes into $x^2$ once. So we put that as our first "digit" of the quotient:

                            1
_______________________________
x^2+x+2       |    x^2


Now, multiply $1$ by $x^2+x+2$, and subtract it from the dividend

                            1
_______________________________
x^2+x+2       |    x^2
- (x^2 + x + 2)
---------------
- x - 2


Now, look at the leading term of what's left: $-x$; and the leading term of the divisor: $x^2$. How many times does $x^2$ go into $-x$? Zero times. So we're done with the division: the quotient is $1$, the remainder is $-x-2$. So we have that

$$\frac{x^2}{x^2+x+2} = 1 + \frac{-x-2}{x^2+x+2} = 1 - \frac{x+2}{x^2+x+2}.$$

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Woo-hoo! Just what I was looking for! Thank you, Arturo! – Oliver Spryn Mar 15 '12 at 4:11

Since $x^2 - 1 \cdot (x^2 + x + 2) = -x - 2$ (one step of polynomial division), we get $x^2 : (x^2 + x + 2) = 1$, remainder $-x-2$, or equivalently, $$\frac{x^2}{x^2 + x + 2} = 1 + \frac{-x-2}{x^2 + x + 2}.$$ This is, in fact, the final result, since the degree of $-x-2$ is less than that of $x^2 + x + 2$.

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Hint: try expressing the fraction as $$\frac{x^2}{x^2+x+2} = \frac{\quad x^2\quad+(x+2)\quad-(x+2)\quad}{x^2+x+2}.$$

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