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.