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:

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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.

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Plug in our p:

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

Multiply:

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=

Let's go back to the problem:

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Let's distribute that 3:

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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

< 4, -5>

step-by-step explanation:

starting from g. we move 4 right and then 5 down to get to h. that means the vector is < 4, -5>

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step-by-step explanation: