Questions: For the polynomial function f(x)=-x^2(x-2)(x+5), answer parts a through e. b. Find the x-intercept(s). The x-intercept(s) is/are -5.0,0.0,2.0. At which zero(s) does the graph of the function cross the x-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The graph crosses the x-axis at the zero(s) B. There are no zeros at which the graph crosses the x-axis.

For the polynomial function f(x)=-x^2(x-2)(x+5), answer parts a through e.
b. Find the x-intercept(s).

The x-intercept(s) is/are -5.0,0.0,2.0.
At which zero(s) does the graph of the function cross the x-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The graph crosses the x-axis at the zero(s)
B. There are no zeros at which the graph crosses the x-axis.

Transcript text: For the polynomial function $f(x)=-x^{2}(x-2)(x+5)$, answer parts a through $e$. b. Find the $x$-intercept(s). The $x$-intercept(s) is/are $-5.0,0.0,2.0$. At which zero(s) does the graph of the function cross the x -axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The graph crosses the $x$-axis at the zero(s) $\square$ B. There are no zeros at which the graph crosses the $x$-axis.

Solution

Solution Steps

Solution Approach

To find the x-intercepts of the polynomial function $ f(x) = -x^2(x-2)(x+5) $, we need to determine the values of $ x $ for which $ f(x) = 0 $. This involves setting the polynomial equal to zero and solving for $ x $. The x-intercepts occur at the roots of the equation. Additionally, to determine at which zeros the graph crosses the x-axis, we need to analyze the multiplicity of each root. If a root has an odd multiplicity, the graph crosses the x-axis at that root.

Step 1: Determine the End Behavior of the Polynomial

The polynomial function given is $ f(x) = -x^2(x-2)(x+5) $.

To determine the end behavior, we need to consider the leading term of the polynomial when expanded. The leading term is determined by multiplying the highest degree terms from each factor:

\[ -x^2 \cdot x \cdot x = -x^4 \]

Since the leading term is $-x^4$, which is a negative coefficient with an even degree, the end behavior of the graph is:

As $ x \to -\infty $, $ f(x) \to -\infty $
As $ x \to \infty $, $ f(x) \to -\infty $

Thus, the graph of $ f(x) $ falls to the left and falls to the right.

Step 2: Find the $ x $-Intercepts

The $ x $-intercepts of the function are found by setting $ f(x) = 0 $:

\[ -x^2(x-2)(x+5) = 0 \]

This equation is satisfied when any of the factors is zero:

$ -x^2 = 0 $ gives $ x = 0 $
$ x - 2 = 0 $ gives $ x = 2 $
$ x + 5 = 0 $ gives $ x = -5 $

Thus, the $ x $-intercepts are $ x = 0, 2, -5 $.

Step 3: Determine Where the Graph Crosses the $ x $-Axis

To determine where the graph crosses the $ x $-axis, we need to consider the multiplicity of each zero:

$ x = 0 $ has a multiplicity of 2 (even), so the graph touches the $ x $-axis and turns around.
$ x = 2 $ has a multiplicity of 1 (odd), so the graph crosses the $ x $-axis.
$ x = -5 $ has a multiplicity of 1 (odd), so the graph crosses the $ x $-axis.

Therefore, the graph crosses the $ x $-axis at $ x = 2 $ and $ x = -5 $.

Final Answer

The end behavior of the graph is: The graph of $ f(x) $ falls to the left and falls to the right.
The $ x $-intercepts are: $\boxed{-5, 0, 2}$
The graph crosses the $ x $-axis at: $\boxed{-5, 2}$

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