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

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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.
A polynomial with a double root at $$-2$$ and another root at $$5$$ is: $$ f(x) = (x + 2)^2(x - 5) $$ Expanding it gives: $$ f(x) = x^3 - x^2 - 16x - 20 $$ You can adjust the leading coefficient $$a$$ to scale the polynomial as desired.

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

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