Question

select the zeros of $h(x) = x^4 + x^3 - 6x^2$.
identify the graph of the function.

Explanation:

Step1: Factor out common term

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

Step2: Factor quadratic polynomial

$h(x) = x^2(x+3)(x-2)$

Step3: Solve for zeros (set $h(x)=0$)

$x^2=0 \implies x=0$; $x+3=0 \implies x=-3$; $x-2=0 \implies x=2$

Step4: Analyze end behavior & shape

Leading term $x^4$, so ends go up. Double root at $x=0$ (graph touches x-axis), single roots at $x=-3,2$ (crosses x-axis).

Answer:

Zeros: $x=-3$, $x=0$, $x=2$
Graph: The bottom-right graph (touches x-axis at $x=0$, crosses at $x=-3$ and $x=2$, ends pointing upward)