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Division of one polynomial by another one.

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Division and exponentiation are two basic operations.

Q: Which operation between two polynomials will not always result in a polynomial?

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The sum of two polynomials is always a polynomial. Therefore, it follows that the sum of more than two polynomials is also a polynomial.

No. Even if the answer is zero, zero is still a polynomial.

Clouser

Closure

That property is called CLOSURE.

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The sum of two polynomials is always a polynomial. Therefore, it follows that the sum of more than two polynomials is also a polynomial.

No. Even if the answer is zero, zero is still a polynomial.

yes

Yes. A polynomial multiplying by a polynomial will always have a multi-termed product. Hope this helps!

Clouser

It is called the property of "closure".

Closure

A zero of the derivative will always appear between two zeroes of the polynomial. However, they do not always alternate. Sometimes two or more zeroes of the derivative will occur between two zeroes of a polynomial. This is often seen with quartic or quintic polynomials (polynomials with the highest exponent of 4th or 5th power).

That property is called CLOSURE.

Let's try an example:The difference between (6x3 + x2 - 4x + 9) and (6x3 + x2 - 4x + 7) is 2 .2 is a polynomial of degree 0, so this example would appear to support the hypothesis in the question.However, polynomials cannot include negative exponents. So, (2x)/(2x2) produces 1/x, which is not a polynomial.So no, not always.

Yes. Note that specifically, the sum might be a constant (just a number), or even zero, but it is convenient to include those in the definition of "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.