Question

1. which could be the function of the following graph?

\\(f(x) = (x + 5)^3(x + 1)^2(x - 2)^2\\)

\\(f(x) = (x + 5)^2(x + 1)(x - 2)^3\\)

\\(f(x) = (x + 5)^3(x + 1)^2(x - 2)^2(x - 5)\\)

\\(f(x) = (x + 5)(x + 1)^3(x - 2)^2\\)

Explanation:

Identify the x-intercepts

Using the End Behavior of Polynomials knowledge point, we first locate where the graph crosses or touches the \(x\)-axis.
Looking at the grid, the horizontal axis has tick marks spaced by \(1\) unit.
The graph meets the \(x\)-axis at three distinct points:

• \(x = -5\)
• \(x = -1\)
• \(x = 2\)

Thus, the corresponding factors of the polynomial must be of the form \((x + 5)\), \((x + 1)\), and \((x - 2)\).

Analyze root multiplicities

Using the Leading Coefficient Test and behavior at intercepts:

• At \(x = -5\), the graph touches the \(x\)-axis and turns around (tangent). This indicates an even multiplicity:

$$ \text{Multiplicity of } (x + 5) \text{ is even (e.g., 2).} $$

• At \(x = -1\), the graph crosses directly through the \(x\)-axis. This indicates an odd multiplicity, typically \(1\):

$$ \text{Multiplicity of } (x + 1) \text{ is odd (e.g., 1 or 3).} $$

• At \(x = 2\), the graph crosses the \(x\)-axis with an inflection-like flattening (wiggle). This indicates an odd multiplicity greater than \(1\):

$$ \text{Multiplicity of } (x - 2) \text{ is odd and } \ge 3 \text{ (e.g., 3).} $$

Match with the given options

We evaluate the given choices to find the one matching these multiplicities:

• Option 1: \(f(x) = (x+5)^3(x+1)^2(x-2)^2\) (Incorrect: even multiplicity at \(x=-5\))
• Option 2: \(f(x) = (x+5)^2(x+1)(x-2)^3\) (Correct: even multiplicity at \(-5\), multiplicity \(1\) at \(-1\), multiplicity \(3\) at \(2\))
• Option 3: \(f(x) = (x+5)^3(x+1)^2(x-2)^2(x-5)\) (Incorrect: has an extra root at \(x=5\))
• Option 4: \(f(x) = (x+5)(x+1)^3(x-2)^2\) (Incorrect: odd multiplicity at \(x=-5\))

Verify end behavior

Using the Leading Coefficient Test

$$
LATEXBLOCK0
$$

This matches the graph, which points upwards on both the far left and far right.

Answer:

• (A) \(f(x) = (x + 5)^3(x + 1)^2(x - 2)^2\)
• (B) \(f(x) = (x + 5)^2(x + 1)(x - 2)^3\) (Correct answer)
• (C) \(f(x) = (x + 5)^3(x + 1)^2(x - 2)^2(x - 5)\)
• (D) \(f(x) = (x + 5)(x + 1)^3(x - 2)^2\)