Find a degree 3 polynomial having zeros -1, 3 and 4 and leading coefficient equal to 1. You can give your answer in factored form.

The polynomial is

To find a degree 3 polynomial with zeros at \( x = -1 \), \( x = 3 \), and \( x = 4 \) and a leading coefficient of 1, we can start with the factored form of the polynomial based on its roots.

If the polynomial has roots \( x = -1 \), \( x = 3 \), and \( x = 4 \), then it can be written as:

\[
f(x) = a(x + 1)(x - 3)(x - 4)
\]

Since the leading coefficient is 1, we set \( a = 1 \). This gives:

\[
f(x) = (x + 1)(x - 3)(x - 4)
\]

Thus, the polynomial in factored form is:

\[
f(x) = (x + 1)(x - 3)(x - 4)
\]

Would you like me to expand it or go over any additional steps?

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Here are five questions to expand on this:

1. What is the polynomial in its expanded form?
2. How can we verify the roots of a polynomial?
3. How would the polynomial change if the leading coefficient was not 1?
4. What is the sum of the roots of the polynomial?
5. How does the factored form help in determining the polynomial's graph?

**Tip:** For a polynomial, the sum of the roots is equal to the negative of the coefficient of the \(x^{n-1}\) term divided by the leading coefficient.

Find a degree 3 polynomial having zeros -1, 3 and 4 and leading coefficient equal to 1. You can give your answer in factored form.

Discover how to construct a degree 3 polynomial with specified zeros and a leading coefficient of 1. This solution provides a step-by-step approach to finding the polynomial in factored form, covering essential algebraic concepts suitable for Grades 9-12.

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