1. The problem is to understand key properties and rules about polynomial functions and their graphs. 2. A zero of multiplicity $k$ means $(x - c)^k$ divides $f(x)$ but $(x - c)^{k+1}$ does not. 3. Rational Root Theorem: If $\frac{p}{q}$ is a zero in lowest terms, then $p$ divides $a_0$ and $q$ divides $a_n$. 4. Intermediate Value Theorem for polynomials: If $f(a)$ and $f(b)$ have opposite signs, there is at least one zero between $a$ and $b$. 5. A polynomial of degree $n$ has at most $n-1$ turning points and at most $n$ x-intercepts. 6. Polynomial graphs are smooth and continuous with no sharp corners. 7. Polynomial operations include sum, difference, product, and quotient (if denominator nonzero). 8. Polynomial division: For $f(x)$ and $d(x)$ with $\deg(d) \leq \deg(f)$, there exist unique $q(x)$ and $r(x)$ such that $$f(x) = q(x)d(x) + r(x)$$ with $r(x) = 0$ or $\deg(r) < \deg(d)$. 9. Remainder Theorem: Dividing $f(x)$ by $x-c$ leaves remainder $f(c)$. 10. If $p(c) = 0$, then $c$ is a root or zero of $p(x)$. 11. Factor Theorem: $x-c$ is a factor of $f(x)$ if and only if $f(c) = 0$. 12. The graph described is a smooth, continuous curve crossing the x-axis multiple times, consistent with polynomial behavior. This summary reviews fundamental polynomial concepts and their graphical interpretations.