Question

In Problems 41– 48, form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.

$Zeros:-2,2,3;degree3$

Short answer

The polynomial function is$f\left(x\right)=x^3-3x^2-4x+12$

Step-by-step solution

Step 1. Given Information

The zeros of the polynomial$-2,2,3$and degree is 3

Step 2. Forming a Polynomial Function

If $r$is a real zero of a polynomial function $f,$then $x-r$is a factor of $f.$This means that$x-\left(-2\right)=\left(x+2\right),\left(x-2\right),\left(x-3\right)$are factor of$f.$As a result, any polynomial function of the form$f\left(x\right)=a\left(x+2\right)\left(x-2\right)\left(x-3\right)$

Step 3. Solving polynomial function

$$\begin{gathered} f\left(x\right)=\left(x+2\right)\left(x-2\right)\left(x-3\right) \\ f\left(x\right)=\left(x^2-4\right)\left(x-3\right) \\ f\left(x\right)=x^3-3x^2-4x+12 \end{gathered}$$