Solve the polynomial inequality algebraically. x^2-2x-8 ≤ 0
x^2-2x-8
(x-4)(x+2)<=0
x<=4, x<=-2
if (x-4)(x+2)<=0
then x ≥ -2 and x ≤ 4
or
-2 ≤ x ≤ 4
To solve the polynomial inequality algebraically, we need to find the values of x that satisfy the inequality x^2 - 2x - 8 ≤ 0.
Step 1: First, let's find the roots (or zeros) of the polynomial equation x^2 - 2x - 8 = 0.
To do this, we can factorize or use the quadratic formula.
Factoring:
The equation can be factored as (x - 4)(x + 2) = 0.
Setting each factor equal to zero, we get two solutions:
x - 4 = 0. So, x = 4.
x + 2 = 0. So, x = -2.
Step 2: Now, let's plot these values on a number line.
-∞ -2 4 ∞
——————|—————|—————|—————|—————
Step 3: We have divided the number line into three parts: (-∞, -2), (-2, 4), and (4, ∞).
Step 4: Now, let's test a value within each of these intervals to see if it satisfies the inequality. We can choose any value within each interval, for example, -3, 0, and 5.
For the interval (-∞, -2):
Let's substitute x = -3 into the inequality:
(-3)^2 - 2(-3) - 8 ≤ 0
9 + 6 - 8 ≤ 0
7 ≤ 0
Since this is not true, (-∞, -2) is not part of the solution.
For the interval (-2, 4):
Let's substitute x = 0 into the inequality:
(0)^2 - 2(0) - 8 ≤ 0
0 - 0 - 8 ≤ 0
-8 ≤ 0
Since this is true, (-2, 4) is part of the solution.
For the interval (4, ∞):
Let's substitute x = 5 into the inequality:
(5)^2 - 2(5) - 8 ≤ 0
25 - 10 - 8 ≤ 0
7 ≤ 0
Since this is not true, (4, ∞) is not part of the solution.
Step 5: Combine the intervals that satisfy the inequality.
From testing the intervals, we found that the solution to the inequality x^2 - 2x - 8 ≤ 0 is (-2, 4].
So, the algebraic solution to the polynomial inequality x^2 - 2x - 8 ≤ 0 is (-2, 4].