To find the polynomial [tex]\( P(x) \)[/tex] of degree 3 that meets the given requirements, we will follow a step-by-step approach:
1. Identify the Roots and their Multiplicities:
- The polynomial has a root at [tex]\( x = 2 \)[/tex] with multiplicity 2.
- The polynomial has a root at [tex]\( x = -3 \)[/tex] with multiplicity 1.
2. Construct the Polynomial from its Roots:
- A root [tex]\( x = 2 \)[/tex] with multiplicity 2 gives us the factor [tex]\( (x - 2)^2 \)[/tex].
- A root [tex]\( x = -3 \)[/tex] with multiplicity 1 gives us the factor [tex]\( (x + 3) \)[/tex].
3. Form the Polynomial Expression:
- Combining these factors, the polynomial [tex]\( P(x) \)[/tex] can be expressed as:
[tex]\[ P(x) = k \cdot (x - 2)^2 \cdot (x + 3) \][/tex]
Here, [tex]\( k \)[/tex] is a constant that we need to determine.
4. Determine the Leading Coefficient ([tex]\( k \)[/tex]) using the \textit{y}-intercept:
- The [tex]\( y \)[/tex]-intercept is given as [tex]\( y = -3.6 \)[/tex].
- This occurs when [tex]\( x = 0 \)[/tex], therefore, [tex]\( P(0) = -3.6 \)[/tex].
5. Evaluate [tex]\( P(0) \)[/tex] using the Polynomial Expression:
- Substitute [tex]\( x = 0 \)[/tex] into [tex]\( P(x) \)[/tex]:
[tex]\[ P(0) = k \cdot (0 - 2)^2 \cdot (0 + 3) \][/tex]
- Simplify the expression:
[tex]\[ P(0) = k \cdot 4 \cdot 3 = 12k \][/tex]
6. Solve for [tex]\( k \)[/tex]:
- Given [tex]\( P(0) = -3.6 \)[/tex]:
[tex]\[ 12k = -3.6 \][/tex]
- Solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{-3.6}{12} = -0.3 \][/tex]
7. Write the Final Polynomial:
- Substitute [tex]\( k = -0.3 \)[/tex] back into the polynomial expression:
[tex]\[ P(x) = -0.3 \cdot (x - 2)^2 \cdot (x + 3) \][/tex]
Thus, the formula for the polynomial [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = -0.3 (x - 2)^2 (x + 3) \][/tex]
