https://file.aiquickdraw.com/mathbot/file/1727280153229tdzi5j9l.jpg

The image provides a cubic function in standard form: 

\[
f(x) = x^3 - 2x^2 - 7x + 2
\]

Let's analyze the components step by step:

### 1. **Factored Form**: 
We need to factor the cubic polynomial to express the function in its factored form. We can use techniques like synthetic division or the Rational Root Theorem to determine the roots.

The factored form of \( f(x) = x^3 - 2x^2 - 7x + 2 \) is:

\[
f(x) = (x - 2)(x + 1)(x - 1)
\]

### 2. **End Behavior**:
The end behavior of a cubic function is determined by the leading term \(x^3\). As \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \).

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

### 3. **Roots (with multiplicity)**:
From the factored form \( (x - 2)(x + 1)(x - 1) \), the roots are:
- \( x = 2 \) (with multiplicity 1)
- \( x = -1 \) (with multiplicity 1)
- \( x = 1 \) (with multiplicity 1)

### 4. **Domain and Range**:
- **Domain**: Since this is a cubic function, the domain is all real numbers: 
  \[
  (-\infty, \infty)
  \]
  
- **Range**: The range is also all real numbers because a cubic function can take any value from \( -\infty \) to \( \infty \):
  \[
  (-\infty, \infty)
  \]

### 5. **Graph Analysis**:
The provided graph matches the behavior of a cubic function. It crosses the x-axis at \( x = -1 \), \( x = 1 \), and \( x = 2 \), confirming the roots. The general shape of the cubic function also aligns with the end behavior described.

Would you like to explore any particular detail further?

Here are five related questions:
1. How do you determine the end behavior of polynomial functions?
2. What is the Rational Root Theorem, and how does it help in factoring polynomials?
3. How can we confirm the multiplicity of a root using the graph?
4. What is the relationship between the degree of a function and its range?
5. How do you find turning points in cubic functions?

**Tip**: To easily factor polynomials, start by checking possible rational roots using the constant and leading coefficient values!

Analyze the cubic function f(x) = x^3 - 2x^2 - 7x + 2, including its factored form, end behavior, roots, domain, range, and graph.

Explore the cubic function f(x) = x^3 - 2x^2 - 7x + 2. Learn about its factored form, end behavior, roots, and more. Includes a graph and step-by-step explanation for high school algebra students.

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