Polynomial Roots 8B4562

1. **Stating the problem:** We are given the function $$f(x) = (x + 1)^3 \cdot (x - 4) \cdot (x + 5)^3$$ and we want to understand its properties such as roots, multiplicities, and behavior. 2. **Formula and important rules:** The function is a product of factors raised to powers. The roots of the function are the values of $$x$$ that make any factor zero. 3. **Finding roots and their multiplicities:** - From $$ (x + 1)^3 $$, root is $$x = -1$$ with multiplicity 3. - From $$ (x - 4) $$, root is $$x = 4$$ with multiplicity 1. - From $$ (x + 5)^3 $$, root is $$x = -5$$ with multiplicity 3. 4. **Behavior at roots:** - Roots with odd multiplicity (like 1 or 3) cross the x-axis. - Roots with even multiplicity touch the x-axis but do not cross it. 5. **Summary:** - Roots: $$x = -5, -1, 4$$ - Multiplicities: 3, 3, and 1 respectively - The function crosses the x-axis at all these roots because all multiplicities are odd. This analysis helps understand the graph and behavior of $$f(x)$$.