$$f ( x ) = \frac { 1 } { 3 } x ^ { 2 } \left( x ^ { 2 } - 5 \right) ( x - 3 )$$
A) 0 , multiplicity 2 , touches $x$-axis;
3 , multiplicity 1 , crosses $x$-axis;
$\sqrt { 5 }$, multiplicity 1 , crosses $x$-axis
$- \sqrt { 5 }$, multiplicity 1 , crosses $x$-axis
B) 0 , multiplicity 2 , crosses $x$-axis;
3 , multiplicity 1 , touches $x$-axis;
$\sqrt { 5 }$, multiplicity 1 , touches $x$-axis;
$- \sqrt { 5 }$, multiplicity 1 , touches $x$-axis
C) 0 , multiplicity 2, crosses x-axis;
3 , multiplicity 1 , touches $x$-axis
D) 0 , multiplicity 2 , touches $x$-axis;
3 , multiplicity 1 , crosses $x$-axis

Question

$$f ( x ) = \frac { 1 } { 3 } x ^ { 2 } \left( x ^ { 2 } - 5 \right) ( x - 3 )$$
A) 0 , multiplicity 2 , touches $x$-axis;
3 , multiplicity 1 , crosses $x$-axis;
$\sqrt { 5 }$, multiplicity 1 , crosses $x$-axis
$- \sqrt { 5 }$, multiplicity 1 , crosses $x$-axis 
B) 0 , multiplicity 2 , crosses $x$-axis;
 3 , multiplicity 1 , touches $x$-axis;
$\sqrt { 5 }$, multiplicity 1 , touches $x$-axis; 
$- \sqrt { 5 }$, multiplicity 1 , touches $x$-axis
C) 0 , multiplicity 2, crosses x-axis;
 3 , multiplicity 1 , touches $x$-axis
D) 0 , multiplicity 2 , touches $x$-axis; 
3 , multiplicity 1 , crosses $x$-axis

Quizplus · Accepted Answer

The function $f(x) = \frac{1}{3}x^2(x^2 - 5)(x - 3)$ has zeros at $x = 0$, $x = 3$, $x = \sqrt{5}$, and $x = -\sqrt{5}$. The zero at $x = 0$ has multiplicity 2 because $x^2$ is squared, indicating the graph touches the x-axis at this point. The zeros at $x = 3$, $x = \sqrt{5}$, and $x = -\sqrt{5}$ each have multiplicity 1, meaning the graph crosses the x-axis at these points.

f(x)=13x2(x2−5)(x−3)f ( x ) = \frac { 1 } { 3 } x ^ { 2 } \left( x ^ { 2 } - 5 \right) ( x - 3 )f(x)=31​x2(x2−5)(x−3)

$f ( x ) = \frac { 1 } { 3 } x ^ { 2 } \left( x ^ { 2 } - 5 \right) ( x - 3 )$