Question

write a polynomial function ( f ) of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. write the function in standard form. (-5, -1, 2) ( f(x) = square)

Explanation:

Step1: Form linear factors from zeros

If $x=-5$, $x=-1$, $x=2$ are zeros, the factors are $(x+5)$, $(x+1)$, $(x-2)$.

Step2: Multiply the first two factors

Multiply $(x+5)(x+1)$:

$$\begin{align*} (x+5)(x+1)&=x^2 + x + 5x + 5\\ &=x^2 + 6x + 5 \end{align*}$$

Step3: Multiply result by third factor

Multiply $(x^2 + 6x + 5)(x-2)$:

$$\begin{align*} &x(x^2 + 6x + 5) -2(x^2 + 6x + 5)\\ &=x^3 + 6x^2 + 5x -2x^2 -12x -10 \end{align*}$$

Step4: Combine like terms

Combine similar terms to get standard form:
$x^3 + (6x^2-2x^2) + (5x-12x) -10$

Answer:

$x^3 + 4x^2 -7x -10$