For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at
each x -intercept.
-$$f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 3 \right) ( x - 3 )$$
A) 0 , multiplicity 4 , touches $x$-axis;
3 , multiplicity 1 , touches $x$-axis
$\sqrt { 3 }$, multiplicity 1 , crosses $x$-axis;
$- \sqrt { 3 }$, multiplicity 1 , crosses $x$-axis

B) 0, multiplicity 4 , crosses $x$-axis;
3 , multiplicity 1 , crosses $x$-axis;

C) 0 , multiplicity 4 , crosses $x$-axis;
3 , multiplicity 1 , touches $x$-axis;
$\sqrt { 3 }$, multiplicity 1 , touches $x$-axis;
$- \sqrt { 3 }$, multiplicity 1 , touches $x$-axis

D) 0 , multiplicity 4 , touches $x$-axis;
3 , multiplicity 1 , crosses $x$-axis

Question

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at
each x -intercept.
-$$f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 3 \right) ( x - 3 )$$
A) 0 , multiplicity 4 , touches $x$-axis;
3 , multiplicity 1 , touches $x$-axis
$\sqrt { 3 }$, multiplicity 1 , crosses $x$-axis;
$- \sqrt { 3 }$, multiplicity 1 , crosses $x$-axis

B) 0, multiplicity 4 , crosses $x$-axis;
3 , multiplicity 1 , crosses $x$-axis;

C) 0 , multiplicity 4 , crosses $x$-axis;
3 , multiplicity 1 , touches $x$-axis;
$\sqrt { 3 }$, multiplicity 1 , touches $x$-axis;
$- \sqrt { 3 }$, multiplicity 1 , touches $x$-axis

D) 0 , multiplicity 4 , touches $x$-axis;
3 , multiplicity 1 , crosses $x$-axis

Quizplus · Accepted Answer

The polynomial has real zeros at $x = 0$ with multiplicity 4, indicating the graph touches the x-axis at this point because the multiplicity is even. It also has a real zero at $x = 3$ with multiplicity 1, meaning the graph crosses the x-axis since the multiplicity is odd. Additionally, it has zeros at $x = \sqrt{3}$ and $x = -\sqrt{3}$, both with multiplicity 1, indicating the graph crosses the x-axis at these points as well.

For the Polynomial, List Each Real Zero and Its Multiplicity f(x)=15x4(x2−3)(x−3)f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 3 \right) ( x - 3 )f(x)=51​x4(x2−3)(x−3)

For the Polynomial, List Each Real Zero and Its Multiplicity $f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 3 \right) ( x - 3 )$