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

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

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

D) 0 , multiplicity 4 , 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.
-$$f ( x ) = \frac { 1 } { 5 } x ^ { 4 } \left( x ^ { 2 } - 3 \right)$$
A) 0 , multiplicity 4 , touches $x$-axis

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

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

D) 0 , multiplicity 4 , crosses $x$-axis

Quizplus · Accepted Answer

The polynomial can be factored as $x^4(x^2 - 3)$. The zero at $x = 0$ has multiplicity 4, indicating the graph touches the x-axis at this point. The zeros at $x = \sqrt{3}$ and $x = -\sqrt{3}$ each have multiplicity 1, indicating the graph crosses the x-axis at these points.

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

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