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.
-$$f ( x ) = 5 \left( x ^ { 2 } + 2 \right) ( x + 1 ) ^ { 2 }$$
A) $- 2$, multiplicity 1 , crosses the $x$-axis; $- 1$, multiplicity 2 , crosses the $x$-axis
B) $- 2$, multiplicity 1 , crosses the $x$-axis; $- 1$, multiplicity 2 , touches the $x$-axis and turns around.
C) $- 1$, multiplicity 2 , touches the $x$-axis and turns around
D) $- 1$, multiplicity 2 , crosses the $x$-axis

Question

Quizplus · Accepted Answer

The polynomial $f(x) = 5(x^2 + 2)(x + 1)^2$ has no real zeros from the factor $x^2 + 2$ because $x^2 + 2 = 0$ has no real solutions. The factor $(x + 1)^2$ gives a zero at $x = -1$ with multiplicity 2. Since the multiplicity is even, the graph touches the x-axis and turns around at $x = -1$.

Find the Zeros for the Polynomial Function and Give the Multiplicity