Question

Find a polynomial f(x) of degree 4 that has the following zeros.
• 5, -3, 7, -4
Leave your answer in factored form.

Answer 1

To find a polynomial \( f(x) \) of degree 4 that has the given zeros \( 5, -3, 7, \) and \( -4 \), you can express the polynomial in factored form using these zeros.

The general form of a polynomial based on its zeros \( r_1, r_2, r_3, r_4 \) is given by:

\[
f(x) = k(x - r_1)(x - r_2)(x - r_3)(x - r_4)
\]

Here, \( k \) is a constant (which we typically set to 1 for a monic polynomial). The roots provided are:

- \( r_1 = 5 \)
- \( r_2 = -3 \)
- \( r_3 = 7 \)
- \( r_4 = -4 \)

Thus, the polynomial \( f(x) \) can be written as:

\[
f(x) = (x - 5)(x + 3)(x - 7)(x + 4)
\]

This is the polynomial in factored form with the specified zeros.