Questions: Q4 Problem 3: Multiple Choice 2 Points Which of the following describes the end behavior of p(x)=-(x+2)^2(x-1)(x-2) "up-up", that is y → ∞ as x → -∞ and y → ∞ as x → ∞ "down-down", that is y → -∞ as x → -∞ and y → -∞ as x → ∞ "up-down", that is y → ∞ as x → -∞ and y → -∞ as x → ∞ "down-up", that is y → -∞ as x → -∞ and y → ∞ as x → ∞

Q4 Problem 3: Multiple Choice
2 Points
Which of the following describes the end behavior of p(x)=-(x+2)^2(x-1)(x-2)
"up-up", that is y → ∞ as x → -∞ and y → ∞ as x → ∞
"down-down", that is y → -∞ as x → -∞ and y → -∞ as x → ∞
"up-down", that is y → ∞ as x → -∞ and y → -∞ as x → ∞
"down-up", that is y → -∞ as x → -∞ and y → ∞ as x → ∞

Transcript text: Q4 Problem 3: Multiple Choice 2 Points Which of the following describes the end behavior of $p(x)=-(x+2)^{2}(x-1)(x-2)$ "up-up", that is $y \rightarrow \infty$ as $x \rightarrow-\infty$ and $y \rightarrow \infty$ as $x \rightarrow \infty$ "down-down", that is $y \rightarrow-\infty$ as $x \rightarrow-\infty$ and $y \rightarrow-\infty$ as $x \rightarrow \infty$ "up-down", that is $y \rightarrow \infty$ as $x \rightarrow-\infty$ and $y \rightarrow-\infty$ as $x \rightarrow \infty$ "down-up", that is $y \rightarrow-\infty$ as $x \rightarrow-\infty$ and $y \rightarrow \infty$ as $x \rightarrow \infty$

Solution

Solution Steps

To determine the end behavior of the polynomial $ p(x) = -(x+2)^2(x-1)(x-2) $, we need to consider the leading term when the polynomial is expanded. The leading term will dominate the behavior of the polynomial as $ x $ approaches $ \infty $ or $ -\infty $.

Identify the degree of the polynomial by adding the exponents of each factor.
Determine the sign of the leading coefficient by considering the product of the coefficients of each factor.
Use the degree and the sign of the leading coefficient to determine the end behavior.

Step 1: Expand the Polynomial

We start with the polynomial $ p(x) = -(x + 2)^2(x - 1)(x - 2) $. Upon expanding this expression, we find: \[ p(x) = -x^4 - x^3 + 6x^2 + 4x - 8 \]

Step 2: Identify the Leading Term

The leading term of the polynomial is the term with the highest degree, which is: \[ \text{Leading term} = -x^4 \]

Step 3: Determine the Degree and Leading Coefficient

The degree of the polynomial is $ 4 $, and the leading coefficient is $ -1 $.

Step 4: Analyze End Behavior

Since the leading coefficient is negative and the degree is even, the end behavior of the polynomial is: