Question

which statement describes the graph of this polynomial function?

\\(f(x) = x^4 + x^3 - 2x^2\\)

• the graph crosses the \\(x\\)-axis at \\(x = 2\\) and \\(x = -1\\) and touches the \\(x\\)-axis at \\(x = 0\\).
• the graph touches the \\(x\\)-axis at \\(x = 2\\) and \\(x = -1\\) and crosses the \\(x\\)-axis at \\(x = 0\\).
• the graph crosses the \\(x\\)-axis at \\(x = -2\\) and \\(x = 1\\) and touches the \\(x\\)-axis at \\(x = 0\\).
• the graph touches the \\(x\\)-axis at \\(x = -2\\) and \\(x = 1\\) and crosses the \\(x\\)-axis at \\(x = 0\\).

Explanation:

Factor the polynomial function

$$ f(x) = x^4 + x^3 - 2x^2 = x^2(x^2 + x - 2) = x^2(x + 2)(x - 1) $$

Determine the roots and their multiplicities

The roots of \(f(x) = 0\) are:

• \(x = 0\) with multiplicity \(2\)
• \(x = -2\) with multiplicity \(1\)
• \(x = 1\) with multiplicity \(1\)

Analyze the behavior at each root

• At \(x = 0\) (even multiplicity of \(2\)), the graph touches the \(x\)-axis.
• At \(x = -2\) and \(x = 1\) (odd multiplicity of \(1\)), the graph crosses the \(x\)-axis.

Answer:

• The graph crosses the x-axis at \(x = 2\) and \(x = -1\) and touches the x-axis at \(x = 0\).
• The graph touches the x-axis at \(x = 2\) and \(x = -1\) and crosses the x-axis at \(x = 0\).
• The graph crosses the x-axis at \(x = -2\) and \(x = 1\) and touches the x-axis at \(x = 0\). (Correct answer)
• The graph touches the x-axis at \(x = -2\) and \(x = 1\) and crosses the x-axis at \(x = 0\).