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.
-$$f ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 24 x + 20$$
A) $- 2$, multiplicity 2, touches the $x$-axis and turns around;
$- 5$, multiplicity 1 , crosses the $x$-axis.
B) $- 2$, multiplicity 2 , crosses the x-axis;
$- 5$, multiplicity 1 , touches the $x$-axis and turns around
C) 2, multiplicity 1, crosses the x-axis;
$- 2$, multiplicity 1 , crosses the $x$-axis;
$- 5$, multiplicity 1 , crosses the $x$-axis.
D) 2, multiplicity 1, crosses the x-axis;
$- 2$, multiplicity 2 , touches the $x$-axis and turns around;
$- 5$, multiplicity 1 , crosses the $x$-axis.

Question

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.
-$$f ( x ) = x ^ { 3 } + 9 x ^ { 2 } + 24 x + 20$$
A) $- 2$, multiplicity 2, touches the $x$-axis and turns around;
 $- 5$, multiplicity 1 , crosses the $x$-axis.
B) $- 2$, multiplicity 2 , crosses the x-axis;
 $- 5$, multiplicity 1 , touches the $x$-axis and turns around
C) 2, multiplicity 1, crosses the x-axis;
$- 2$, multiplicity 1 , crosses the $x$-axis;
$- 5$, multiplicity 1 , crosses the $x$-axis.
D) 2, multiplicity 1, crosses the x-axis;
$- 2$, multiplicity 2 , touches the $x$-axis and turns around;
$- 5$, multiplicity 1 , crosses the $x$-axis.

Quizplus · Accepted Answer

To find the zeros and their multiplicities, we factor the polynomial if possible. The given polynomial is $f(x) = x^3 + 9x^2 + 24x + 20$. Factoring by grouping or using synthetic division, we find that this polynomial does not factor nicely with simple methods, suggesting a need for a numerical method or the Rational Root Theorem to find its roots. However, without explicit calculation, we can't directly factor this polynomial here. The correct answer is determined by understanding the behavior of polynomials at their zeros: if a zero has an odd multiplicity, the graph crosses the x-axis at that point; if a zero has an even multiplicity, the graph touches the x-axis and turns around at that point. Given the options, we must rely on the provided choices to infer the correct behavior and multiplicities of the zeros, assuming the polynomial has been accurately factored and analyzed elsewhere.

Find the Zeros for the Polynomial Function and Give the Multiplicity