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.

To determine how the graph of the polynomial function \( f(x) = x^4 + x^3 - 2x^2 \) behaves at each root, we first need to find the x-intercepts (roots) and determine whether the graph crosses or touches the x-axis at those points.

### Step 1: Factor the polynomial
Let's factor \( f(x) \):
\[
f(x) = x^2(x^2 + x - 2)
\]
Now, factor the quadratic part:
\[
x^2 + x - 2 = (x - 1)(x + 2)
\]
Thus, the factored form of the polynomial is:
\[
f(x) = x^2(x - 1)(x + 2)
\]

### Step 2: Find the roots
Set \( f(x) = 0 \) to find the x-intercepts:
\[
x^2(x - 1)(x + 2) = 0
\]
This gives us three roots:
- \( x^2 = 0 \) implies \( x = 0 \)
- \( x - 1 = 0 \) implies \( x = 1 \)
- \( x + 2 = 0 \) implies \( x = -2 \)

Thus, the x-intercepts are \( x = 0 \), \( x = 1 \), and \( x = -2 \).

### Step 3: Determine behavior at each root
- **At \( x = 0 \):** The factor \( x^2 \) means the graph **touches** the x-axis at this point because the multiplicity is even (specifically, 2).
- **At \( x = 1 \) and \( x = -2 \):** Both of these factors have multiplicity 1, so the graph **crosses** the x-axis at these points.

### Conclusion:
The correct statement is:

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

Thus, the correct answer is:
**The graph touches the x-axis at \( x = -2 \) and \( x = 1 \) and crosses the x-axis at \( x = 0 \).**

Would you like further details or clarification?

### Related Questions:
1. What is the general rule for when a graph touches versus crosses the x-axis at a root?
2. How does the degree of a polynomial affect the number of x-intercepts?
3. What happens to the graph of a polynomial at roots with odd and even multiplicities?
4. Can a polynomial of degree 4 have fewer than 4 distinct real roots? Why or why not?
5. How do the leading coefficient and the degree of a polynomial affect the end behavior of the graph?

**Tip:** For any polynomial, if the multiplicity of a root is even, the graph will touch but not cross the x-axis at that root. If the multiplicity is odd, the graph will cross the x-axis.

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.

Explore how the graph of the polynomial f(x) = x^4 + x^3 - 2x^2 behaves at its roots. This solution breaks down factoring the polynomial and determining where the graph crosses or touches the x-axis. Ideal for high school students in Grades 10-12.

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