Answer:
Case (1): P(x) + Q(x) will always be a polynomial function.
Case (2): P(x) . Q(x) will always be a polynomial function.
Case (3): P(x) / Q(x) may sometimes be a polynomial function depending upon the fact whether the value of Q(x) is equal to zero OR Q(x) is not equal to zero.
But, P(x) / Q(x) will always be a polynomial function provided that Q(x) [tex]\neq 0[/tex]
But, if Q(x) = 0, then polynomial function will never be a polynomial function as it will become undefined.
Case (4): 1 / Q(x) will never be a polynomial function because we cannot have the variable appearing as the denominator of a function as A polynomial function is a function involving only non-negative (positive and 0) integer powers of 'x'.
Step-by-step explanation:
Consider the polynomial functions P (x) and Q (x).
Case 1: P(x) + Q(x)
P(x) + Q(x) will always be a polynomial function.
lets suppose,
P(x) = 2x and Q(x) = 5x
P(x) + Q(x) = 2x + 5x = 7x
Case 2: P(x) . Q(x)
P(x) . Q(x) will always be a polynomial function.
lets suppose,
P(x) = 2x and Q(x) = 5x
P(x) + Q(x) = 2x × 5x = 10[tex]x^{2}[/tex]
Case 3: P(x) / Q(x)
We can say P(x) / Q(x) may sometimes be a polynomial function depending upon the fact whether the value of Q(x) is equal to zero OR Q(x) is not equal to zero.
But, P(x) / Q(x) will always be a polynomial function provided that Q(x) [tex]\neq 0[/tex]
lets suppose,
P(x) = 2x and Q(x) = 5x
P(x) / Q(x) = 2x / 5x = 2/5
But, if Q(x) = 0, then polynomial function will never be a polynomial function as it will become undefined.
lets suppose,
P(x) = 2x and Q(x) = 0
P(x) / Q(x) = 2x / 0 = Undefined
So, we can say P(x) / Q(x) may sometimes be a polynomial function depending upon the fact whether the value of Q(x) is equal to zero OR Q(x) is not equal to zero.
Case 4: 1 / Q(x)
1 / Q(x) will never be a polynomial function because we cannot have the variable appearing as the denominator of a function as A polynomial function is a function involving only non-negative (positive and 0) integer powers of 'x'.
lets suppose,
Q(x) = 5x
1 / Q(x) = 1 / 5x = [tex]x^{-1}[/tex]/5
As polynomial function contains negative integer powers of 'x'. So, it can never be a polynomial function.
Learn more about polynomial functions from brainly.com/question/10881257
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