Question

find the zeros of $f(x)=x^3 + x^2 - 6x$.
$x=-4$ $x=-1$ $x=2$
$x=-3$ $x=0$ $x=3$
$x=-2$ $x=1$ $x=8$
identify the graph of the function
four graphs of cubic functions are shown, with options to choose from

Explanation:

Step1: Factor out common term

$F(x) = x(x^2 + x - 6)$

Step2: Factor quadratic polynomial

$x^2 + x - 6 = (x+3)(x-2)$, so $F(x) = x(x+3)(x-2)$

Step3: Solve for zeros

Set $F(x)=0$:
$x=0$, $x+3=0 \implies x=-3$, $x-2=0 \implies x=2$

Step4: Analyze end behavior

Leading term is $x^3$, as $x\to+\infty$, $F(x)\to+\infty$; as $x\to-\infty$, $F(x)\to-\infty$.

Step5: Match zeros and end behavior

The graph with zeros at $x=-3,0,2$ and correct end behavior is the top-right graph.

Answer:

Zeros: $x=-3$, $x=0$, $x=2$
Graph: The top-right graph (opening upward on the right, downward on the left, crossing x-axis at -3, 0, 2)