Finding polynomials of the lowest possible degree that satisfy a given condition

Hello there,

I need to find the polynomials $\displaystyle f(x)$ and $\displaystyle g(x)$, which are of the *lowest possible degree* and *satisfy the given condition*:

$\displaystyle (x^4-2x^3-4x^2+6x+1) \cdot f(x) + (x^3-5x-3) \cdot g(x) = x^4$

All I want to know is how to find **the lowest possible degree** that the polynomials should be of. I can solve the rest using the method of undetermined coefficients—but if there's another way (preferably easier), then feel free to share.

This is my first post here, so please bear with me if I did anything wrong.

Thanks in advance.

Re: Finding polynomials of the lowest possible degree that satisfy a given condition

Quote:

Originally Posted by

**LJoe21** Hello there,

I need to find the polynomials $\displaystyle f(x)$ and $\displaystyle g(x)$, which are of the *lowest possible degree* and *satisfy the given condition*:

$\displaystyle (x^4-2x^3-4x^2+6x+1) \cdot f(x) + (x^3-5x-3) \cdot g(x) = x^4$

All I want to know is how to find **the lowest possible degree** that the polynomials should be of. I can solve the rest using the method of undetermined coefficients—but if there's another way (preferably easier), then feel free to share.

This is my first post here, so please bear with me if I did anything wrong.

Thanks in advance.

note that the right side of the equation is degree 4. since the two products are summed, that means both products on the left side would have to be 4th degree, also.

f(x) is multiplying a 4th degree poly ... so, wouldn't f(x) have to have degree 0 (i.e. , a constant) ?

so, what degree would g(x) have to be to form a product that is also 4th degree?

Re: Finding polynomials of the lowest possible degree that satisfy a given condition

Well, the solutions are (according to my workbook):

$\displaystyle f(x) = 9x^2 - 26x - 21$

$\displaystyle g(x) = -9x^3 + 44x^2 - 39x - 7$

and I've absolutely no idea how to determine that $\displaystyle f(x)$ should be of second degree and $\displaystyle g(x)$ of third.

Re: Finding polynomials of the lowest possible degree that satisfy a given condition

This has nothing to do with your original question. Your original question asked for the **lowest** degree of polynomials that would satisfy the given equation. The answer given in your second post are NOT of lowest degree.