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Q: After you divide a polynomial by a monomial you can check your answer by multiplying it by the original?

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You multiplay the consents and who cares about math, go out and party! the world will end in 2012 anyway!To divide a polynomial by a monomial you don't need to multiply the polynomial.

You divide each term of the binomial by the monomial, and add everything up. This also works for the division of any polynomial by a monomial.

If the quotient of a certain binomial and 20x2 is is the polynomial

You use long division of polynomials.

That means that you divide one polynomial by another polynomial. Basically, if you have polynomials "A" and "B", you look for a polynomial "C" and a remainder "R", such that: B x C + R = A ... such that the remainder has a lower degree than polynomial "B", the polynomial by which you are dividing. For example, if you divide by a polynomial of degree 3, the remainder must be of degree 2 or less.

Determine the GCF .If it is 1 then continue with the next step but if it is a number such as three then remove that number and divide each monomial by that number and put the polynomial within a set of parentheses with the GCF on the outside of the parentheses

For example, if you divide a polynomial of degree 2 by a polynomial of degree 1, you'll get a result of degree 1. Similarly, you can divide a polynomial of degree 4 by one of degree 2, a polynomial of degree 6 by one of degree 3, etc.

multiplied

That's the definition of a "rational function". You simply divide a polynomial by another polynomial. The result is called a "rational function".

You can find explanation and examples here: http://en.wikipedia.org/wiki/Polynomial_division

Yes. If you add, subtract or multiply (but not if you divide) any two polynomials, you will get a polynomial.

It means that you can do any of those operations, and again get a number from the set - in this case, a polynomial. Note that if you divide a polynomial by another polynomial, you will NOT always get a polynomial, so the set of polynomials is not closed under division.