50 ! ! consider the polynomial functions p (x) and q (x) , where neither polynomial is a constant function.

determine whether each of the following related functions below will be polynomial function. fill in the blank with always, sometimes, or never for each related function.

p(x) + q(x)

p(x) · q(x)

p(x) / q(x)

1 / p(x)

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.

a.)the line for function b is steeper