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*f*(*x*) = *x*2(*x* – *a*)2(*x* – *b*)4(*x* – *c*)

*f*(*x*) = *x*3(*x* – *a*)3(*x* – *b*)(*x* – *c*)2

*f*(*x*) = *x*4(*x* – *a*)(*x* – *b*)3(*x* – *c*)3

*f*(*x*) = (*x* – *a*)2(*x* – *b*)(*x* – *c*)6

To determine which function could represent the graph, let’s analyze the key characteristics of the graph, especially the zeroes, multiplicities, and general behavior.

### Observations from the Graph
1. **Roots**: The graph crosses the x-axis at least three distinct points, indicating at least three distinct roots.
2. **Multiplicity of Roots**: The graph shows some roots with an odd multiplicity (it crosses the x-axis) and some with an even multiplicity (it touches the x-axis without crossing it).
3. **Degree of Polynomial**: The overall shape suggests a higher degree polynomial with positive leading behavior (as \( x \to \infty \), \( f(x) \to \infty \)).

### Analyzing Each Option
Let’s break down each given function to match these characteristics:

1. **Option 1**: \( f(x) = x^2(x - a)^2(x - b)^4(x - c) \)
   - Degree: \( 2 + 2 + 4 + 1 = 9 \)
   - Roots: \( x = 0, x = a, x = b, x = c \)
   - The graph of a degree-9 polynomial would generally end with opposite signs as \( x \to \pm \infty \). However, this graph ends with both ends up, so this option may not fit.

2. **Option 2**: \( f(x) = x^3(x - a)^3(x - b)(x - c)^2 \)
   - Degree: \( 3 + 3 + 1 + 2 = 9 \)
   - Roots: \( x = 0, x = a, x = b, x = c \)
   - Similar to Option 1, this has a degree-9, but with a similar issue of sign behavior at both ends.

3. **Option 3**: \( f(x) = x^4(x - a)(x - b)^3(x - c)^3 \)
   - Degree: \( 4 + 1 + 3 + 3 = 11 \)
   - Roots: \( x = 0, x = a, x = b, x = c \)
   - Degree-11 would also show opposite end behavior, which doesn’t match.

4. **Option 4**: \( f(x) = (x - a)^2(x - b)(x - c)^6 \)
   - Degree: \( 2 + 1 + 6 = 9 \)
   - Roots: \( x = a, x = b, x = c \) only, no \( x = 0 \).
   - Matches an odd multiplicity for root

Let a, b, and c be real numbers where a ≠ b ≠ c ≠ 0. Which of the following functions could represent the graph below?

f(x) = x^2(x - a)^2(x - b)^4(x - c)
f(x) = x^3(x - a)^3(x - b)(x - c)^2
f(x) = x^4(x - a)(x - b)^3(x - c)^3
f(x) = (x - a)^2(x - b)(x - c)^6

Explore how to determine which polynomial function best represents a given graph based on its roots, multiplicity, and end behavior. The problem provides functions to analyze, focusing on concepts such as polynomial degrees, multiplicity of roots, and end behavior. Suitable for advanced high school math students.

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