Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the
x-axis or touches the x-axis and turns around, at each zero.
-$$f ( x ) = \left( x + \frac { 1 } { 5 } \right) ^ { 4 } \left( x ^ { 2 } + 7 \right) ^ { 5 }$$
A) $- \frac { 1 } { 5 }$, multiplicity 4 , crosses the $x$-axis.
B) $- \frac { 1 } { 5 }$, multiplicity 4 , touches the $x$-axis and turns around; $- 7$, multiplicity 5 , crosses the $x$-axis
C) $\frac { 1 } { 5 }$, multiplicity 4 , touches the $x$-axis and turns around; 7 , multiplicity 5 , crosses the $x$-axis
D) $- \frac { 1 } { 5 }$, multiplicity 4 , touches the $x$-axis and turns around.

Question

Quizplus · Accepted Answer

The polynomial has a zero at $-\frac{1}{5}$ with multiplicity 4, which is an even number, indicating the graph touches the x-axis and turns around at this point. The term $(x^2 + 7)^5$ does not produce real zeros because $x^2 + 7 > 0$ for all real numbers $x$, so there are no additional zeros to consider.

Find the Zeros for the Polynomial Function and Give the Multiplicity