Question

identify the zeros of $p(x) = x^6 - 11x^5 + 30x^4$.
identify the graph of the function.

Explanation:

Step1: Factor out common term

$p(x) = x^4(x^2 - 11x + 30)$

Step2: Factor quadratic expression

$x^2 - 11x + 30 = (x-5)(x-6)$, so $p(x) = x^4(x-5)(x-6)$

Step3: Solve for zeros

Set $p(x)=0$:
$x^4=0 \implies x=0$;
$x-5=0 \implies x=5$;
$x-6=0 \implies x=6$

Step4: Analyze graph behavior

• The leading term $x^6$ means as $x\to\pm\infty$, $p(x)\to+\infty$.
• $x=0$ is a zero with even multiplicity (4), so the graph touches the x-axis here and turns around.
• $x=5$ and $x=6$ are zeros with odd multiplicity (1), so the graph crosses the x-axis here.
• The graph will have a "flat" touch at $x=0$, then cross at $x=5$ and $x=6$, matching the bottom-left graph.

Answer:

Zeros: $x=0$, $x=5$, $x=6$
Graph: The bottom-left graph (the one that touches the x-axis at $x=0$, crosses at positive x-values, and rises to $+\infty$ as $x\to\pm\infty$)