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Problem:

Let $$f(x) = 3\frac{x^4+x^3+x^2+1}{x^2+x-2}.$$Give a polynomial $g(x)$ so that $f(x) + g(x)$ has a horizontal asymptote of $0$ as $x$ approaches positive infinity.

How would I start this? I found the partial fractions expansion of this, but what do I do next?

mucciolo
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JenkinsMa
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    Long division of numerator by denominator to write $f(x)$ in the form $q(x) + \frac{r(x)}{x^2+x-2}$, where $r(x)$ has degree $< 2$. Then as $x \to \infty$, $r(x)/(x^2+x-2) \to 0$, so the polynomial you want is $-q(x)$. – Robert Israel Apr 26 '17 at 21:33

1 Answers1

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Let $a(x)=3x^4+3x^3+3x^2+3$ and $b(x)=x^2+x-2$ so that $f(x)=\frac{a(x)}{b(x)}$. Since we are dealing with a division of polynomials, there exist two other polynomials $q$ and $r$, such that the degree of $r$ is less than the degree of $b$ and $a=qb+r$, or equivalently $$ \frac{a}{b}=q+\frac{r}{b}. $$ Now, since $\deg r<\deg b$, when $x\to+\infty$ we have that $\frac{r}{b}\to 0$, therefore $f$ is asymptotically like $q$ at infinity. You should be able to find what needs to be added in order to obtain what you want, now.

yellon
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