Question

determine the end behavior of the polynomial function.
y = 2(x + 3)^2(x - 5)

a. as x \to -\infty, f(x) \to -\infty and x \to \infty,
f(x) \to -\infty.
b. as x \to -\infty, f(x) \to \infty and x \to \infty,
f(x) \to -\infty.
c. as x \to -\infty, f(x) \to -\infty and x \to \infty,
f(x) \to \infty.
d. as x \to -\infty, f(x) \to \infty and x \to \infty, f(x) \to \infty.

Explanation:

Step1: Find leading term degree

Expand the highest-power terms:
$(x+3)^2$ has leading term $x^2$, multiplied by $(x-5)$ gives $x^3$, then multiplied by 2 gives $2x^3$. Degree is 3 (odd).

Step2: Identify leading coefficient

Leading coefficient of $2x^3$ is $2$ (positive).

Step3: Apply end behavior rules

For odd degree, positive leading coefficient:
As $x \to -\infty$, $f(x) \to -\infty$; as $x \to \infty$, $f(x) \to \infty$.

Answer:

c. As $x \to -\infty, f(x) \to -\infty$ and $x \to \infty, f(x) \to \infty$.