Questions: Find a polynomial (f(x)) of degree 5 that has the following zeros: (e, -2) multiplicatively 2, (3, -4). Leave your answer in factored form: (r(x)=)

Transcript text: Find a polynomial $f(x)$ of degree 5 that has the following zeros. \[ \text { e, }-2 \text { imuttiptictly 2), 3, }-4 \] Leave your answer in factored form. \[ r(x)= \]

Solution

Solution Steps

Step 1: Identify the Zeros

The problem requires us to find a polynomial $ f(x) $ of degree 5 with the specified zeros: $ e, -2, 3, -4 $. Since $ e $ is a real number and $ -2 $ is mentioned as a zero with multiplicity 2, we can list the zeros as follows:

$ e $
$ -2 $ (with multiplicity 2)
$ 3 $
$ -4 $

Step 2: Construct the Polynomial

To construct the polynomial, we use the factored form based on the zeros. The polynomial can be expressed as: \[ f(x) = (x - e)(x + 2)^2(x - 3)(x + 4) \]

Step 3: Simplify the Polynomial

The polynomial can be expanded and simplified, but since the problem asks for the polynomial in factored form, we will keep it as: \[ f(x) = (x - e)(x + 2)^2(x - 3)(x + 4) \]