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

Hello,

Step-by-step explanation:

Explanation:

+

We need p(x) need to be a degree 2 polynomial so the numerator of the second fraction is degree 4. Our goal is to cancel the terms of the first fraction's numerator that are of degree 2 or higher.

So let p(x)=ax^2+bx+c.

+

Plug in our p:

+

Take a break to multiply the factors of our second fraction's numerator.

Multiply:

=

+

+

=

Let's go back to the problem:

+

Let's distribute that 3:

+.

Next we have . Based on previous statement this forces .

Next we have . With and , this gives .

So which equals .

Lastly, .

This makes .

This implies or simplified