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.

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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:

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

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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 $a$:

$a = \frac{7}{-3} = -\frac{7}{3}$

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Step 4: Write the final polynomial function.

Now that we know $a = -\frac{7}{3}$, the polynomial function is:

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

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Step 5: Expand if needed (optional).

You can leave it in the factored form above, or distribute $-\frac{7}{3}$ if necessary:

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

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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?

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