Write the equation of the polynomial function of least degree possible with the zeros −1, 1, 3; and passing through the point 
(2, 7).

To determine the **polynomial function** of the least degree with given zeros \(x = -1\), \(x = 1\), and \(x = 3\), and that passes through the point \((2, 7)\), we follow these steps:

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### Step 1: Write the general form of the polynomial function.
The **roots** \(x = -1\), \(x = 1\), and \(x = 3\) tell us that the polynomial will have the factors:

\[
P(x) = a(x + 1)(x - 1)(x - 3)
\]

where \(a\) is a constant that we need to determine.

---

### Step 2: Simplify the polynomial.
First, expand the product \((x + 1)(x - 1)\):

\[
(x + 1)(x - 1) = x^2 - 1
\]

Now multiply by the remaining factor \((x - 3)\):

\[
(x^2 - 1)(x - 3) = x^3 - 3x^2 - x + 3
\]

Thus, the polynomial in terms of \(a\) is:

\[
P(x) = a \left( x^3 - 3x^2 - x + 3 \right)
\]

---

### Step 3: Use the point \((2, 7)\) to find the value of \(a\).
We are given that the polynomial passes through the point \((2, 7)\), meaning when \(x = 2\), \(P(x) = 7\). Substitute \(x = 2\) and \(P(x) = 7\) into the equation:

\[
7 = a \left( 2^3 - 3(2^2) - 2 + 3 \right)
\]

Simplify the expression inside the parentheses:

\[
2^3 = 8, \quad 3(2^2) = 12
\]
\[
8 - 12 - 2 + 3 = -3
\]

Thus:

\[
7 = a(-3)
\]

Solve for \(a\):

\[
a = \frac{7}{-3} = -\frac{7}{3}
\]

---

### Step 4: Write the final polynomial function.
Now that we know \(a = -\frac{7}{3}\), the polynomial function is:

\[
P(x) = -\frac{7}{3} \left( x^3 - 3x^2 - x + 3 \right)
\]

---

### Step 5: Expand if needed (optional).
You can leave it in the factored form above, or distribute \( -\frac{7}{3} \) if necessary:

\[
P(x) = -\frac{7}{3}x^3 + 7x^2 + \frac{7}{3}x - 7
\]

---

Would you like further clarifications or any other details?

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### 5 Follow-up Questions:
1. How do you determine the degree of a polynomial given its zeros?
2. What would the polynomial look like if it passed through a different point, such as \((0, 5)\)?
3. What is the impact of changing the sign of the coefficient \(a\) in a polynomial function?
4. How do multiplicities of roots affect the shape of the graph?
5. Can a polynomial function have non-real roots? How would that change the factorization?

---

**Tip:** Always double-check whether the polynomial is of the least degree by confirming that all provided roots are accounted for, without extra factors.

Write the equation of the polynomial function of least degree possible with the zeros −1, 1, 3; and passing through the point (2, 7).

Learn how to find the polynomial function of least degree with given zeros −1, 1, 3, and that passes through the point (2, 7). This problem explores polynomial roots, algebraic factorization, and determining constants using known points. Suitable for students in Grades 10-12.

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