1. **Problem Statement:** Determine whether the related functions formed from polynomials $P(x)$ and $Q(x)$ (neither constant) are polynomial functions, choosing from Always, Sometimes, or Never. 2. **Recall:** A polynomial function is a function of the form $$a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$ where $n$ is a non-negative integer and coefficients $a_i$ are constants. 3. **Analyze each related function:** - **$P(x) + Q(x)$:** - Sum of two polynomials is always a polynomial. - So, this is **Always** a polynomial. - **$P(x) \cdot Q(x)$:** - Product of two polynomials is always a polynomial. - So, this is **Always** a polynomial. - **$\frac{P(x)}{Q(x)}$:** - Division of polynomials is not always a polynomial. - It is a polynomial only if $Q(x)$ divides $P(x)$ exactly (no remainder). - So, this is **Sometimes** a polynomial. - **$\frac{1}{P(x)}$:** - Reciprocal of a polynomial is generally not a polynomial. - Only if $P(x)$ is a nonzero constant (which is excluded here), it would be polynomial. - Since $P(x)$ is not constant, this is **Never** a polynomial. 4. **Summary:** | Related functions | Always | Sometimes | Never | |-------------------|--------|-----------|-------| | $P(x)+Q(x)$ | O | | | | $P(x)\cdot Q(x)$ | O | | | | $\frac{P(x)}{Q(x)}$| | O | | | $\frac{1}{P(x)}$ | | | O | **Final answers:** - $P(x)+Q(x)$: Always - $P(x)\cdot Q(x)$: Always - $\frac{P(x)}{Q(x)}$: Sometimes - $\frac{1}{P(x)}$: Never