Well, first off, @issrabi's polynomial long division example above is a bit confusing... at first I think it was wrong, but it's just that the problem he DID is different than the problem that he STATED.
He stated (x³- 2x² + x – 1)/( x- 2)= the quotient 3x² +4x+ 9 and reminder 17
But he meant to say (3x³- 2x² + x – 1)/( x- 2)= the quotient 3x² +4x+ 9 and reminder 17
As first stated, of course, you get a diffferent result... but in any case, I would not do either by long division, I would use synthetic division (sooooo much easier)....
... but that's neither here not there, since that has nothing do with your problem:
\(\Large \text{Create a unique example of dividing a polynomial}\)
\(\Large \color{red}{ \text{ by a monomial}} \text{ and provide the simplified form.}\)
x - 2 is not a monomial, it is a binomial.
A monomial is a one-term polynomial, and every monomial IS a polynomial (but not every polynomial is a monomial). A polynomial is just a sum of terms that have coefficients and non-negative integer exponents (which can include a constant term).
so:
\(x^4+3x^2+5x-4\) is a polynomial (with 4 terms)
\(3x^2+5x-4\) is a polynomial (with 3 terms, called a trinomial)
\(3x^2-1\) is a polynomial (with 2 terms, called a binomial)
\(5x^6\) is a monomial, which is simply a polynomial with 1 term
So, I suppose, your question is asking you to do something like:
\[\Large \dfrac{ 4x^4+6x^3-8x^2+4x }{ 2x }\]
That is "dividing a polynomial by a monomial ".... But I'm honestly not sure what it is getting at when it says " the two ways used to simplify this expression"??
I guess you might say that one way it to factor the GCF from the numerator and cancel, and another way is to break it up into a sum of individual terms, and then reduce each term?? Eg:
\[\Large \dfrac{ 4x^4+6x^3-8x^2+4x }{ 2x }= \dfrac{ 4x^4}{ 2x }+ \dfrac{ 6x^3}{ 2x }- \dfrac{ 8x^2 }{ 2x }+ \dfrac{4x }{ 2x }\]
????....... not sure.