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To find a polynomial function \( f(x) \) with zeros at \( x = -3 \), \( x = 1 \), and \( x = 5 \), and which also passes through the point \( (7, 360) \), follow these steps:

### Step 1: Set up the Polynomial Form
Since the zeros of the polynomial are given, we can express \( f(x) \) as:
\[
f(x) = a(x + 3)(x - 1)(x - 5)
\]
where \( a \) is a constant that we'll determine using the given point \( (7, 360) \).

### Step 2: Substitute the Point \( (7, 360) \)
Since \( f(7) = 360 \), we substitute \( x = 7 \) and \( f(x) = 360 \) into the equation:
\[
360 = a(7 + 3)(7 - 1)(7 - 5)
\]

### Step 3: Simplify the Expression
Calculate each factor inside the parentheses:
\[
360 = a(10)(6)(2)
\]
\[
360 = a \cdot 120
\]

### Step 4: Solve for \( a \)
Divide both sides by 120 to isolate \( a \):
\[
a = \frac{360}{120} = 3
\]

### Step 5: Write the Final Polynomial
Now that we know \( a = 3 \), we can write the polynomial as:
\[
f(x) = 3(x + 3)(x - 1)(x - 5)
\]

### Step 6: Expand the Polynomial (Optional)
Expanding this gives:
\[
f(x) = 3(x^3 - 3x^2 - 13x + 15) = 3x^3 - 9x^2 - 39x + 45
\]

So, the polynomial function is:
\[
f(x) = 3x^3 - 9x^2 - 39x + 45
\]

Would you like further details or have any questions?

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### Related Questions
1. How do we determine the degree of a polynomial based on its zeros?
2. What does the value of \( a \) signify in the polynomial \( f(x) = a(x + 3)(x - 1)(x - 5) \)?
3. How would the polynomial change if it had a zero with multiplicity?
4. What are some methods to expand and simplify polynomial expressions?
5. How would we find the polynomial if it passed through a different point, say \( (2, -40) \)?

### Tip
To verify your polynomial, substitute the zeros and the given point back into the polynomial to check consistency.

Find a polynomial function with the zeros -3, 1, 5 whose graph passes through the point (7, 360).

Learn how to find a polynomial function with zeros at -3, 1, and 5, passing through the point (7, 360). This example includes step-by-step solutions, covering concepts like polynomial functions, zeros, and graphing. Suitable for high school students.

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