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

In order to build a polynomial we need one or more terms. A term is a number, variable (denoted by a letter) or any combination of numbers and variables held together by multiplication. The following are examples of terms:

[tex]5, 3x, -5ab c^{2}, \frac{2x^{3}y }{5}, x^{6} [/tex]

Now it might look like one of those involves division but it can be thought of as multiplication by (2/5). When we do this the exponents must be positive.

Polynomials are expressions made up of terms held together by addition and subtraction. Again, the exponents must be positive. Since polynomials are made up of the sum or difference of terms, adding or subtracting polynomials just leads to more polynomials. Here are some examples of Polynomials:

[tex]4xy-3 x^{2} +7, \frac{2x}{5}-3abc-5j [/tex]

Now let’s consider what happens if we multiply polynomials. As an example we use: [tex](x+5)(2x-y)=2 x^{2} +10x-xy-5y[/tex]

What you might notice is that multiplication will lead us to multiply terms (but multiplying terms gives us more term,as) and also to add or subtract terms but that just gives more polynomials. Therefore multiplication leads to more polynomials.

Finally, we consider division. Here a simple example will do the trick: 2 is a term and x is a term. Let us divide 2 by x. We get: [tex] \frac{\2}{x}=2 x^{-1} [/tex]  which is not a polynomial because we have a negative exponent. 

Thus, the answer to your question is division. Division of polynomials will not always result in a polynomial.

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