1. **Problem:** Determine the degree and multiplicity of each root for the polynomial \(h(t) = -3t^3 (t + 2)(t + 5)\). 2. **Step 1: Identify the degree of the polynomial.** - The degree is the sum of the exponents of all factors. - Here, \(t^3\) has degree 3, \((t+2)\) has degree 1, and \((t+5)\) has degree 1. - Total degree = \(3 + 1 + 1 = 5\). 3. **Step 2: Find the roots and their multiplicities.** - Roots are values of \(t\) that make the polynomial zero. - From \(t^3\), root at \(t=0\) with multiplicity 3. - From \(t+2=0\), root at \(t=-2\) with multiplicity 1. - From \(t+5=0\), root at \(t=-5\) with multiplicity 1. 4. **Step 3: Behavior around each root.** - Multiplicity odd (like 3 or 1) means the graph crosses the x-axis at that root. - Multiplicity even means the graph touches and bounces off the x-axis. - Here, all multiplicities are odd, so the graph crosses the x-axis at each root. 5. **Summary:** - Degree: 5 - Roots and multiplicities: \(0\) (3), \(-2\) (1), \(-5\) (1) - Graph crosses x-axis at all roots. \[\boxed{\text{Degree} = 5, \quad \text{Roots: } 0(3), -2(1), -5(1)}\]