To find the real zeros of the polynomial $Q(x) = (x + 5)^2(x - 3)^4$, we'll go through the following steps:

Step 1: Identify the Zeros and Their Multiplicities

The given polynomial is factored as: $Q(x) = (x + 5)^2(x - 3)^4$

• The factor $(x + 5)^2$ gives a zero at $x = -5$ with multiplicity 2.
• The factor $(x - 3)^4$ gives a zero at $x = 3$ with multiplicity 4.

Real Zeros and Multiplicities:

• $x = -5$ with multiplicity 2.
• $x = 3$ with multiplicity 4.

Step 2: Behavior at Each Zero

• For a zero with even multiplicity, the graph touches the x-axis at that zero but does not cross it.
• For a zero with odd multiplicity, the graph crosses the x-axis at that zero.

Zero at $x = -5$:

Since the multiplicity is 2 (even), the graph will touch the x-axis at $x = -5$ but not cross it.

Zero at $x = 3$:

Since the multiplicity is 4 (even), the graph will touch the x-axis at $x = 3$ but not cross it.

Step 3: Sign Chart

We'll examine the sign of $Q(x)$ in the intervals determined by the zeros $x = -5$ and $x = 3$.

• For $x < -5$: Both $(x + 5)^2$ and $(x - 3)^4$ are positive (since even powers are always positive). So, $Q(x) > 0$ for $x < -5$.
• For $-5 < x < 3$: In this interval, $(x + 5)^2 > 0$ and $(x - 3)^4 > 0$, so $Q(x) > 0$.
• For $x > 3$: Both factors are positive, so $Q(x) > 0$ here as well.

Since the polynomial is non-negative in all regions, the graph stays above the x-axis and touches the axis at $x = -5$ and $x = 3$.

Step 4: Rough Sketch of the Graph

• The graph touches the x-axis at $x = -5$ and $x = 3$, without crossing it.
• Since the polynomial is positive for all values of $x$, the graph remains above the x-axis for all intervals.
• The behavior near each zero is parabolic, as the multiplicities are even.

Step 5: Use of a Graphing Utility

You can use a graphing calculator or software (like Desmos or GeoGebra) to confirm the behavior of the graph. The graph should match the following:

• It touches the x-axis at $x = -5$ and $x = 3$.
• The graph is non-negative and remains above the x-axis.

Let me know if you'd like me to help you graph this or provide further details.

Expansion Questions:

1. How does the degree of the polynomial affect the end behavior of the graph?
2. What is the impact of the multiplicity of a zero on the graph of a polynomial?
3. Why does the graph not cross the x-axis at the points where the zeros have even multiplicity?
4. How would the graph change if one of the multiplicities was odd instead of even?
5. How can sign charts help predict the behavior of polynomial graphs?

Tip: When analyzing polynomial graphs, always focus on the zeros and their multiplicities, as they determine whether the graph crosses or touches the x-axis at each zero.