o) Build polynomial functions with a double root at -2 and nother root at 5 .

To construct a polynomial function with a **double root at \(-2\)** and another root at \(5\), you can follow these steps: ### 1. **Understanding Roots and Multiplicity** - A **double root** at \(-2\) means that \((x + 2)\) is a factor **twice**. - A **single root** at \(5\) means that \((x - 5)\) is a factor **once**. ### 2. **Constructing the Polynomial** Combine these factors to form the polynomial: \[ f(x) = a(x + 2)^2(x - 5) \] where \(a\) is any non-zero constant (it determines the leading coefficient and can scale the polynomial vertically). ### 3. **Expanding the Polynomial (Optional)** If you prefer the expanded form, you can multiply the factors: \[ \begin{align*} f(x) &= a(x + 2)^2(x - 5) \\ &= a(x^2 + 4x + 4)(x - 5) \\ &= a\left(x^3 + 4x^2 + 4x - 5x^2 - 20x - 20\right) \\ &= a\left(x^3 - x^2 - 16x - 20\right) \\ &= a x^3 - a x^2 - 16a x - 20a \end{align*} \] ### 4. **Example Polynomial** For simplicity, let’s choose \(a = 1\): \[ f(x) = (x + 2)^2(x - 5) = x^3 - x^2 - 16x - 20 \] ### 5. **General Form** Any polynomial with the specified roots can be written as: \[ f(x) = a(x + 2)^2(x - 5) \quad \text{where} \quad a \neq 0 \] You can choose different values of \(a\) to get different polynomial functions that satisfy the root conditions. ### **Graphical Interpretation** - **Double Root at \(-2\):** The graph of the polynomial will touch the x-axis at \(x = -2\) and turn around, indicating a repeated root. - **Single Root at \(5\):** The graph will cross the x-axis at \(x = 5\). ### **Summary** A suitable polynomial with a double root at \(-2\) and another root at \(5\) is: \[ f(x) = (x + 2)^2(x - 5) \] You can expand it to: \[ f(x) = x^3 - x^2 - 16x - 20 \] Feel free to choose any non-zero constant \(a\) to scale the polynomial as needed.