Given the provided function:

[tex]\[ (x) = (x-3)(x-2)(x+1)^2 \][/tex]

Determine the end behavior of the function.

A. Rises to the left and falls to the right
B. Falls to the left and falls to the right
C. Falls to the left and rises to the right
D. Rises to the left and rises to the right



Answer :

To determine the end behavior of the polynomial \( f(x) = (x-3)(x-2)(x+1)^2 \), we need to analyze its leading term, which dictates the behavior of the polynomial as \( x \) approaches positive and negative infinity.

1. Identify the Highest Degree Terms from Each Factor:
- From \( (x-3) \), the highest degree term is \( x \).
- From \( (x-2) \), the highest degree term is \( x \).
- From \( (x+1)^2 \), the highest degree term is \( (x)^2 = x^2 \).

2. Determine the Leading Term of the Polynomial:
- Multiply the highest degree terms together: \( x \cdot x \cdot x^2 \).
- This gives us the leading term \( x^4 \).

3. Analyze the Leading Term \( x^4 \):
- The degree of \( x^4 \) is 4, which is an even number.
- For even-degree polynomials, the end behavior is that both ends of the graph will go in the same direction.
- The coefficient of \( x^4 \) is positive (1).

4. Determine the End Behavior:
- Since the coefficient is positive and the degree is even:
- As \( x \) approaches positive infinity (\( x \to +\infty \)), \( f(x) \to +\infty \) (the right end rises).
- As \( x \) approaches negative infinity (\( x \to -\infty \)), \( f(x) \to +\infty \) (the left end rises).

Thus, the polynomial \( f(x) = (x-3)(x-2)(x+1)^2 \) rises to the left and rises to the right. The correct answer is:

D) rises to the left and rises to the right