Question

write a polynomial function ( f ) of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros (-2, 1, 3). write the polynomial in standard form. ( f(x) = square )

Explanation:

Step1: Use factor theorem for zeros

If $x=a$ is a zero, then $(x-a)$ is a factor. For zeros $-2,1,3$, the factors are $(x+2)$, $(x-1)$, $(x-3)$.
$f(x)=(x+2)(x-1)(x-3)$

Step2: Multiply first two factors

Expand $(x+2)(x-1)$ using FOIL method.
$(x+2)(x-1)=x^2 - x + 2x - 2 = x^2 + x - 2$

Step3: Multiply result with third factor

Multiply $(x^2 + x - 2)$ by $(x-3)$.

$$\begin{align*} &(x^2 + x - 2)(x-3)\\ =&x^2(x-3) + x(x-3) - 2(x-3)\\ =&x^3 - 3x^2 + x^2 - 3x - 2x + 6\\ =&x^3 - 2x^2 - 5x + 6 \end{align*}$$

Answer:

$x^3 - 2x^2 - 5x + 6$