Question

find a polynomial with the following zeros.
-3, 2, 7
f(x) = x³ + ?x² + x +

Explanation:

Step1: Form factors from zeros

If \( r \) is a zero of a polynomial, then \( (x - r) \) is a factor. For zeros \(-3\), \(2\), \(7\), the factors are \( (x + 3) \), \( (x - 2) \), \( (x - 7) \).

Step2: Multiply the factors

First, multiply \( (x + 3)(x - 2) \):
\[

$$\begin{align*} (x + 3)(x - 2)&=x^2 - 2x + 3x - 6\\ &=x^2 + x - 6 \end{align*}$$

\]
Then multiply the result by \( (x - 7) \):
\[

$$\begin{align*} (x^2 + x - 6)(x - 7)&=x^3 - 7x^2 + x^2 - 7x - 6x + 42\\ &=x^3 - 6x^2 - 13x + 42 \end{align*}$$

\]

Answer:

For the \( x^2 \) term coefficient: \(-6\)
For the \( x \) term coefficient: \(-13\)
For the constant term: \(42\)

(So the polynomial is \( f(x) = x^3 - 6x^2 - 13x + 42 \))