$$f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 5 \right)$$
A) 0 , multiplicity 4 , touches $x$-axis;
$\sqrt { 5 }$, multiplicity 1 , crosses $x$-axis;
$- \sqrt { 5 }$, multiplicity 1 , crosses $x$-axis
B) 0 , multiplicity 4 , crosses $x$-axis
C) 0 , multiplicity 4 , crosses $x$-axis;
$\sqrt { 5 }$, multiplicity 1 , touches $x$-axis;
D) 0 , multiplicity 4 , touches $x$-axis
$- \sqrt { 5 }$, multiplicity 1 , touches $x$-axis

Question

Quizplus · Accepted Answer

The function $f(x) = \frac{1}{5}x^4(x^2 - 5)$ has roots at $x = 0$, $x = \sqrt{5}$, and $x = -\sqrt{5}$. The root at $x = 0$ has multiplicity 4 because $x^4$ is a factor, indicating the graph touches the x-axis at this point. The roots at $x = \sqrt{5}$ and $x = -\sqrt{5}$ each have multiplicity 1, indicating the graph crosses the x-axis at these points.

f(x)=15x4(x2−5)f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 5 \right)f(x)=51​x4(x2−5)

$f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 5 \right)$