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.
-$$f ( x ) = x ^ { 3 } + x ^ { 2 } - 20 x$$
A) 0 , multiplicity 1 , crosses the $x$-axis
- 5, multiplicity 1 , crosses the $x$-axis
4, multiplicity 1 , crosses the $x$-axis
B) 0 , multiplicity 1 , crosses the $x$-axis
5 , multiplicity 1 , crosses the $x$-axis
$- 4$, multiplicity 1 , crosses the $x$-axis
C) 0 , multiplicity 1 , touches the $x$-axis and turns around;
- 5, multiplicity 1 , touches the $x$-axis and turns around;
4 , multiplicity 1 , touches the $x$-axis and turns around
D) $- 5$, multiplicity 2 , touches the $x$-axis and turns around 4 , multiplicity 1 , crosses the $x$-axis

Question

Quizplus · Accepted Answer

The polynomial $f(x) = x^3 + x^2 - 20x$ can be factored as $x(x^2 + x - 20)$, which further factors into $x(x + 5)(x - 4)$. This gives us zeros at $x = 0$, $x = -5$, and $x = 4$, each with multiplicity 1. Since each zero has a multiplicity of 1, which is odd, the graph crosses the $x$-axis at each zero.

Find the Zeros for the Polynomial Function and Give the Multiplicity