For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at
each x -intercept.
-$$f ( x ) = \left( x + \frac { 1 } { 4 } \right) ^ { 2 } \left( x ^ { 2 } + 7 \right) ^ { 3 }$$
A) $- \frac { 1 } { 4 }$, multiplicity 2 , touches $x$-axis
B) $- \frac { 1 } { 4 }$, multiplicity 2 , touches $x$-axis; $- 7$, multiplicity 3 , crosses $x$-axis
C) $- \frac { 1 } { 4 }$, multiplicity 2 , crosses $x$-axis
D) $\frac { 1 } { 4 }$, multiplicity 2 , touches $x$-axis; 7 , multiplicity 3 , crosses $x$-axis

Question

Quizplus · Accepted Answer

The polynomial has a real zero at $x = -\frac{1}{4}$ with multiplicity 2, which means the graph touches the x-axis at this point. The factor $(x^2 + 7)^3$ does not produce real zeros because $x^2 + 7 > 0$ for all real $x$, so there are no x-intercepts associated with this factor.

For the Polynomial, List Each Real Zero and Its Multiplicity f(x)=(x+14)2(x2+7)3f ( x ) = \left( x + \frac { 1 } { 4 } \right) ^ { 2 } \left( x ^ { 2 } + 7 \right) ^ { 3 }f(x)=(x+41​)2(x2+7)3

For the Polynomial, List Each Real Zero and Its Multiplicity $f ( x ) = \left( x + \frac { 1 } { 4 } \right) ^ { 2 } \left( x ^ { 2 } + 7 \right) ^ { 3 }$