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

Quizplus · Accepted Answer

To find the zeros of the polynomial $f(x) = x^3 + 4x^2 - x - 4$, we can attempt to factor it or use synthetic division. Factoring by grouping, we get $x^2(x + 4) - 1(x + 4) = (x^2 - 1)(x + 4) = (x + 1)(x - 1)(x + 4)$. Thus, the zeros are $x = -1$, $x = 1$, 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