To write the polynomial function based on the given points, we will use the x-intercepts and analyze the additional points provided.
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Identify the x-intercepts: From the points given, the x-intercepts are at:
- \( x = -3 \)
- \( x = 0 \)
- \( x = 1 \)
- \( x = 2 \)
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Create factors for each x-intercept:
- For \( x = -3 \), the factor is \( (x + 3) \)
- For \( x = 0 \), the factor is \( x \)
- For \( x = 1 \), the factor is \( (x - 1) \)
- For \( x = 2 \), the factor is \( (x - 2) \)
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Multiply the factors together to form the polynomial: \[ f(x) = k(x + 3)(x)(x - 1)(x - 2) \] where \( k \) is a constant multiplier that we can find using another point from the graph.
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Use another point to find \( k \): We can use the point \( (2.25, 68) \) to determine \( k \).
Substitute \( x = 2.25 \) into the polynomial: \[ f(2.25) = k(2.25 + 3)(2.25)(2.25 - 1)(2.25 - 2) \] Calculate the components:
- \( 2.25 + 3 = 5.25 \)
- \( 2.25 - 1 = 1.25 \)
- \( 2.25 - 2 = 0.25 \)
Now substituting: \[ f(2.25) = k (5.25)(2.25)(1.25)(0.25) = 68 \]
Calculate \( (5.25)(2.25)(1.25)(0.25) \):
- Calculate step by step:
- \( 5.25 \times 2.25 = 11.8125 \)
- \( 11.8125 \times 1.25 = 14.765625 \)
- \( 14.765625 \times 0.25 = 3.69140625 \)
Now we have: \[ k \cdot 3.69140625 = 68 \] Solving for \( k \): \[ k = \frac{68}{3.69140625} \approx 18.41 \]
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Final Polynomial in Factored Form: The polynomial function in factored form is: \[ f(x) \approx 18.41 (x + 3)(x)(x - 1)(x - 2) \]
Thus, the final answer for the polynomial function in factored form is approximately \[ f(x) = 18.41 (x + 3)(x)(x - 1)(x - 2). \]
(Note: If you need \( k \) in its exact form, it can be computed exactly based on the earlier calculations for precise numerical results).