Answer:
Always
Always
Sometimes
Never
Step-by-step explanation:
P(X) + Q(X)
Sum of two polynomials is always a polynomial.
For example, Let P(x) = 3x² and Q(x) = 5x
P(x) + Q(x) = 3x² + 5x
This is again a polynomial.
In other words, we can say that Polynomial is closed under addition.
P(x) . Q(x)
Product of two polynomials is again a polynomial.
Let P(x) = 2x and Q(x) = a constant function, 5
Then the product = 10x, is again a polynomial.
Multiplication of two polynomials is closed.
P(x) / Q(x)
This need not always be a polynomial. When Q(x) = a constant function zero, i.e., Q(x) = 0, then the function is not defined.
But let's say P(x) = 5x² and Q(x) = x.
=
= 5x, a polynomial.
So, is a polynomial sometimes.
We can, say Division is not always closed.
1/Q(x)
This could never be a polynomial. This is not even in the form of a polynomial. So, is never a polynomial.
Hence, the answer.