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.
-$$f ( x ) = x ^ { 3 } + 8 x ^ { 2 } - x - 8$$
A) $- 1$, multiplicity 1 , crosse the $x$-axis;
1 , multiplicity 1 , crosses the $x$-axis;
- 8 , multiplicity 1 , crosses the $x$-axis.
B) 8 , multiplicity 1 , crosses the x-axis;
1 , multiplicity 1 , crosses the $x$-axis;
- 8 , multiplicity 1 , crosses the $x$-axis.
C) 1 , multiplicity 2 , touches the $x$-axis and turns around;
$- 8$, multiplicity 1 , crosses the x-axis.
D) $- 1$, multiplicity 1 , touches the $x$-axis and turns around; 1 , multiplicity 1 , touches the $x$-axis and turns around;
- 8 , multiplicity 1 , touches the $x$-axis and turns around Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and make the degree of the function as small as possible.

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.
-$$f ( x ) = x ^ { 3 } + 8 x ^ { 2 } - x - 8$$
A) $- 1$, multiplicity 1 , crosse the $x$-axis; 
1 , multiplicity 1 , crosses the $x$-axis;
 - 8 , multiplicity 1 , crosses the $x$-axis.
B) 8 , multiplicity 1 , crosses the x-axis; 
1 , multiplicity 1 , crosses the $x$-axis; 
- 8 , multiplicity 1 , crosses the $x$-axis.
C) 1 , multiplicity 2 , touches the $x$-axis and turns around;
 $- 8$, multiplicity 1 , crosses the x-axis.
D) $- 1$, multiplicity 1 , touches the $x$-axis and turns around; 1 , multiplicity 1 , touches the $x$-axis and turns around; 
- 8 , multiplicity 1 , touches the $x$-axis and turns around Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or -1 and make the degree of the function as small as possible.

Quizplus · Accepted Answer

The zeros of the polynomial $f(x) = x^3 + 8x^2 - x - 8$ can be found by factoring or using synthetic division. Factoring by grouping, we get $x^2(x + 8) - 1(x + 8) = (x^2 - 1)(x + 8) = (x + 1)(x - 1)(x + 8)$. Thus, the zeros are $x = -1$, $x = 1$, and $x = -8$, each with multiplicity 1. Since each zero has a multiplicity of 1, the graph crosses the x-axis at each zero.

Find the Zeros for the Polynomial Function and Give the Multiplicity