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
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
