To sketch the graph of a fifth-degree polynomial function with the given properties, let’s break down the essential features and behavior:

Properties of the Polynomial:

1. Fifth Degree Polynomial: A fifth-degree polynomial will have the general form $f(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}$, where the sum of the multiplicities $m_1 + m_2 = 5$. Since the leading coefficient is negative, the polynomial will have a downwards end behavior as $x \to \infty$ and upwards end behavior as $x \to -\infty$.

2. One Real Root with Multiplicity 3: This means the polynomial touches the x-axis at a certain point (let's say $x = r$) and flattens out without crossing at that point.

3. Negative Leading Coefficient: The leading coefficient determines the end behavior. Since it is negative, the polynomial will start from positive infinity as $x \to -\infty$ and go down to negative infinity as $x \to \infty$.

Key Points for the Sketch:

• Root of Multiplicity 3: At $x = r$, the graph will touch the x-axis and flatten out but not cross it.
• End Behavior:
  • As $x \to -\infty$, the graph goes to $+\infty$.
  • As $x \to \infty$, the graph goes to $-\infty$.
• Other Real Roots: Since the polynomial is of degree 5, there will be two additional roots (as multiplicities must add up to 5). These other roots can either be distinct real roots (crossing the x-axis) or complex conjugates, which do not show up as intercepts on the real number plane.

Sketching Strategy:

1. Start by plotting the real root $x = r$ where the graph flattens out.
2. Choose some simple additional roots, say $x = r_1$ and $x = r_2$, where the graph crosses the x-axis.
3. Ensure the graph starts from positive infinity and descends, touching the x-axis at $x = r$, crossing it at $x = r_1$ and $x = r_2$.
4. Ensure the overall shape reflects the polynomial’s degree and negative leading coefficient, descending to negative infinity.

I'll summarize this sketch in text form:

1. Right Tail: The graph heads down to negative infinity as $x \to \infty$.
2. Left Tail: The graph heads up to positive infinity as $x \to -\infty$.
3. Root at $r$: The graph flattens at the root with multiplicity 3.
4. Crosses at two other points: The graph crosses the x-axis twice at the other real roots (if applicable).

Would you like to see an illustrative graph of this function?