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To find a formula for \( P(x) \) given the information:

1. The polynomial \( P(x) \) is of degree 3.
2. It has a root of multiplicity 2 at \( x = 1 \).
3. It has a root of multiplicity 1 at \( x = -5 \).
4. The \( y \)-intercept is \( y = -4 \), which means \( P(0) = -4 \).

### Step-by-Step Solution

1. **Set up the factors of the polynomial:**
   - Since \( P(x) \) has a root of multiplicity 2 at \( x = 1 \), one of its factors is \( (x - 1)^2 \).
   - Since \( P(x) \) has a root of multiplicity 1 at \( x = -5 \), another factor is \( (x + 5) \).
   
   Therefore, we can write:
   \[
   P(x) = a(x - 1)^2(x + 5)
   \]
   where \( a \) is a constant that we need to determine.

2. **Use the \( y \)-intercept to find \( a \):**
   - The \( y \)-intercept is given by \( P(0) = -4 \).
   - Substitute \( x = 0 \) into \( P(x) \) and set it equal to \(-4\):
     \[
     P(0) = a(0 - 1)^2(0 + 5) = -4
     \]
     \[
     P(0) = a \cdot 1 \cdot 5 = -4
     \]
     \[
     5a = -4
     \]
     \[
     a = -\frac{4}{5}
     \]

3. **Write the final formula:**
   Substitute \( a = -\frac{4}{5} \) back into the polynomial expression:
   \[
   P(x) = -\frac{4}{5}(x - 1)^2(x + 5)
   \]

### Final Answer
\[
P(x) = -\frac{4}{5}(x - 1)^2(x + 5)
\]

Would you like further explanation, or do you have any questions?

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Here are 5 related questions to deepen your understanding:

1. How does the multiplicity of a root affect the shape of the polynomial graph at that root?
2. What would happen if the root at \( x = 1 \) had multiplicity 3 instead of 2?
3. How can you determine the leading coefficient's effect on the polynomial’s end behavior?
4. Why is the \( y \)-intercept important for determining the constant \( a \)?
5. How would the formula change if the root at \( x = -5 \) had a multiplicity of 2?

**Tip:** When solving for a constant in a polynomial using the \( y \)-intercept, substitute \( x = 0 \) to simplify calculations.

The polynomial of degree 3, P(x), has a root of multiplicity 2 at x = 1 and a root of multiplicity 1 at x = -5. The y-intercept is y = -4. Find a formula for P(x).

Learn how to find a polynomial formula of degree 3 with given roots and multiplicity. This solution involves algebraic concepts such as roots, multiplicity, and determining the constant using the y-intercept. Ideal for students in Grades 10-12.

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