Question

1. \\(\lim\limits_{x\to -\infty} p(x) = -\infty\\) and \\(\lim\limits_{x\to \infty} p(x) = -\infty\\)

Explanation:

Step1: Identify polynomial end behavior

For a polynomial $p(x) = a_nx^n + ... + a_0$, end behavior depends on leading coefficient $a_n$ and degree $n$.

Step2: Match limits to degree/coefficient

We need $\lim_{x\to-\infty} p(x) = -\infty$ and $\lim_{x\to\infty} p(x) = -\infty$. This requires even degree ($n$ even) and negative leading coefficient ($a_n < 0$).

Step3: Choose a simple example

Select lowest even degree ($n=2$) and negative leading coefficient, e.g., $a_n=-1$.
<Expression>
$p(x) = -x^2$
</Expression>

Step4: Verify end behavior

For $p(x)=-x^2$:
$\lim_{x\to\infty} -x^2 = -\infty$, $\lim_{x\to-\infty} -x^2 = -\infty$

Answer:

A valid function is $\boldsymbol{p(x) = -x^2}$ (any even-degree polynomial with a negative leading coefficient is acceptable, e.g., $p(x)=-2x^4 + 3x^2 -1$ also works)