Step 2: Test points on both sides of each zero and vertical asymptote
The function has zeros when x = 1 and x = -1.
The function has vertical asymptotes when x = 2 and x = -2.
Let's test some values for each interval:
- For x < -2, let's pick x = -3. Substituting -3 into the function gives:
F(-3) = ((-3 - 1)(-3 + 1))/((-3 - 2)(-3 + 2)) = ((-4)(-2))/((-5)(-1)) = (8/5) > 0
- For -2 < x < -1, let's pick x = -1.5. Substituting -1.5 into the function gives:
F(-1.5) = ((-1.5 - 1)(-1.5 + 1))/((-1.5- 2)(-1.5 + 2)) = ((-2.5)(-0.5))/((-3.5)(-0.5)) = (1.25) > 0
- For -1 < x < 1, let's pick x = 0. Substituting 0 into the function gives:
F(0) = ((0 - 1)(0 + 1))/((0 - 2)(0 + 2)) = (-1) / (-4) = 0.25 > 0
- For 1 < x < 2, let's pick x = 1.5. Substituting 1.5 into the function gives:
F(1.5) = ((1.5 - 1)(1.5 + 1))/((1.5 - 2)(1.5 + 2)) = ((0.5)(2.5))/((-0.5)(3.5)) = -0.357 < 0
- For x > 2, let's pick x = 3. Substituting 3 into the function gives:
F(3) = ((3 - 1)(3 + 1))/((3 - 2)(3 + 2)) = (8)(4)/(1)(5) = 6.4 > 0
Step 3: Make the sign chart on a number line.
On the number line, we will have:
- A vertical asymptote at x = -2, sign < 0
- A zero at x = -1, sign > 0
- A vertical asymptote at x = 2, sign < 0
- A zero at x = 1, sign > 0
Using this information, we construct the sign chart:
---(-)--(+)---(-)--(+)---(-)---
This represents the sign of the function F(x) = ((x - 1)(x + 1))/((x - 2)(x + 2)) along the number line.
