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.
-Crosses the $x$-axis at $- 1,0$, and 4 ; lies below the $x$-axis between $- 1$ and 0 ; lies above the $x$-axis between 0 and $4 .$
A) $f ( x ) = - x ^ { 3 } + 3 x ^ { 2 } + 4 x$
B) $f ( x ) = - x ^ { 3 } - 3 x ^ { 2 } + 4 x$
C) $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 4 x$
D) $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 4 x$

Question

Quizplus · Accepted Answer

The zeros of the polynomial function $f(x) = -x^3 + 3x^2 + 4x$ are $x = -1, 0, 4$. Factoring out $x$, we get $f(x) = -x(x + 1)(x - 4)$. The zeros are $x = -1$ (crosses the x-axis), $x = 0$ (crosses the x-axis), and $x = 4$ (crosses the x-axis). The graph lies below the x-axis between $-1$ and $0$, and above the x-axis between $0$ and $4$, matching the given behavior. Each zero has a multiplicity of 1, indicating the graph crosses the x-axis at each zero.

Find the Zeros for the Polynomial Function and Give the Multiplicity