For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at
each x -intercept.
-$$f ( x ) = 3 \left( x ^ { 2 } + 5 \right) ( x + 6 ) ^ { 2 }$$
A) $- 6$, multiplicity 2 , touches $x$-axis
B) $- 5$, multiplicity 1 , touches $x$-axis; $- 6$, multiplicity 2 , crosses $x$-axis
C) $- 6$, multiplicity 2 , crosses $x$-axis
D) $- 5$, multiplicity 1 , crosses $x$-axis; $- 6$, multiplicity 2 , touches $x$-axis

Question

Quizplus · Accepted Answer

The polynomial has a real zero at $x = -6$ with multiplicity 2, indicated by the factor $(x + 6)^2$. There are no real zeros from $x^2 + 5$ since it cannot be zero (it's always positive). A zero with even multiplicity means the graph touches the x-axis at that intercept.

For the Polynomial, List Each Real Zero and Its Multiplicity f(x)=3(x2+5)(x+6)2f ( x ) = 3 \left( x ^ { 2 } + 5 \right) ( x + 6 ) ^ { 2 }f(x)=3(x2+5)(x+6)2

For the Polynomial, List Each Real Zero and Its Multiplicity $f ( x ) = 3 \left( x ^ { 2 } + 5 \right) ( x + 6 ) ^ { 2 }$