Find, if it exists, a polynomial 𝑃(𝑥) such that:

[gloomy garden]

someone can help me? idk how to start

[kindred dragon]

am i missing something, is it not just 5(2x^4 -4x + 1)

or is there not infinite solutions for 10x^4 + any string of lesser degree

looking at the leading coefficient of the greatest degree term

[gloomy garden]

This question asks me:

Is the polynomial 𝑃(𝑥) unique?

[kindred dragon]

No the polynomial is not unique

think here, the rule in this case is that for limits approaching infinity, you can simply look at the leading coefficient of the highest degree terms

in this case, the highest degree term on the bottom is x^4 with a coefficient of 2

thus we need to have a leading coefficient on the top of 2•5 such that (2•5)/2 =5 for our limit

so any polynomial in the numerator of form (2•5)x^4 + ax^3 +bx^2 + cx + d satisfies this form

[kindred dragon]

so long answer short, it is not unique

does that make sense?

Sorry idk latex so my math can look a bit jumbled

[brave ferry]

There's no one solution but a family of solution

[dense solar]

For limit x tends to infinity and the limit equals to 5 the px highest degree must be equal to the degree of denominator. Also the leading coefficient must be 10 as the x^4 coefficient of denominator is 2.. the polynomial will be a family of solutions generalized as 10x^4+bx^3+cx^2+dx+e

How did I predict this??

So basically when u divide the denominator and numerator by x^4

All other terms after applying limit will be 0 as anything/infinity is 0 except the terms with x^4

So u will be left with 10/2

Thats 5