Question

chapter 7:7.3-7.4
score: 0/15 answered: 0/15
question 1
solve the polynomial inequality: (2(x + 4)^{2}(x + 10)(x - 7) > 0)
give your answer in interval notation. enter dne if there is no solution.

Explanation:

Step1: Find the boundary points

Set the polynomial equal to zero to find critical values.

$$2(x + 4)^2(x + 10)(x - 7) = 0 \implies x = -10, -4, 7$$

Step2: Determine interval signs

Test a value in each interval to find the sign.

$$\text{For } x \in (-\infty, -10), \text{ sign is } +$$

$$\text{For } x \in (-10, -4), \text{ sign is } -$$

$$\text{For } x \in (-4, 7), \text{ sign is } -$$

$$\text{For } x \in (7, \infty), \text{ sign is } +$$

Step3: Write the solution

Select intervals where the polynomial is strictly positive.

$$x \in (-\infty, -10) \cup (7, \infty)$$

Answer:

\((-\infty, -10) \cup (7, \infty)\)