1.  Find a polynomial function of degree 3 with zeros  -3, 1, and 2  having y-intercept 16.

To find a polynomial function of degree 3 with the given conditions (zeros \(-3\), \(1\), and \(2\) and a y-intercept at \(y = 16\)), follow these steps:

### Step 1: General form of the polynomial
The general form of a polynomial with zeros \(x = -3\), \(x = 1\), and \(x = 2\) is:
\[
f(x) = a(x + 3)(x - 1)(x - 2)
\]
where \(a\) is a constant that we need to determine.

### Step 2: Use the y-intercept condition
We are told that the y-intercept is \(16\), which means that when \(x = 0\), \(f(x) = 16\). Substitute \(x = 0\) into the polynomial:
\[
f(0) = a(0 + 3)(0 - 1)(0 - 2) = 16
\]
Simplifying:
\[
f(0) = a(3)(-1)(-2) = 16
\]
\[
6a = 16
\]
\[
a = \frac{16}{6} = \frac{8}{3}
\]

### Step 3: Write the final polynomial
Now that we know \(a = \frac{8}{3}\), the polynomial function is:
\[
f(x) = \frac{8}{3}(x + 3)(x - 1)(x - 2)
\]

This is the required polynomial of degree 3 with the specified conditions.

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Would you like more details or have any questions?

### Related Questions:
1. How can you verify if the zeros of a polynomial are correct?
2. What happens to the shape of a cubic function if the leading coefficient is negative?
3. Can a cubic function have fewer than three real zeros? If so, when?
4. How does the constant \(a\) affect the graph of the polynomial?
5. Can a cubic function have complex zeros, and how would you find them?

### Tip:
Always check the leading coefficient and intercepts to ensure a polynomial meets the required conditions for both shape and specific points!

Find a polynomial function of degree 3 with zeros -3, 1, and 2 having y-intercept 16.

Discover how to find a cubic polynomial with zeros at -3, 1, and 2, and a y-intercept at 16. This step-by-step solution covers key algebraic concepts such as polynomial functions and the Factor Theorem. Suitable for high school students in Grades 9-12.

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