Questions: Plot the x-intercepts and make a sign chart that represents the function shown below. f(x)=(x+6)(x+4)^2(x+1)(x-1)

Solution Steps

The x-intercepts are the values of x where f(x) = 0. We can find them by setting each factor of the function equal to zero and solving for x:

• x + 6 = 0 => x = -6
• x + 4 = 0 => x = -4
• x + 1 = 0 => x = -1
• x - 1 = 0 => x = 1

The x-intercepts divide the number line into intervals. We pick a test point in each interval and determine the sign of f(x) at that point.

• x < -6: Test point x = -7. f(-7) is (-)(-)(-)(-) which is positive.
• -6 < x < -4: Test point x = -5. f(-5) is (+)(-)(-)(-) which is negative.
• -4 < x < -1: Test point x = -2. f(-2) is (+)(+)(-)(-) which is positive.
• -1 < x < 1: Test point x = 0. f(0) is (+)(+)(+)(-) which is negative.
• x > 1: Test point x = 2. f(2) is (+)(+)(+)(+) which is positive.

Plot the x-intercepts on a number line and indicate the sign of f(x) in each interval:

      +      -      +      -      +
<-----(-6)----(-4)----(-1)----(1)----->

Final Answer: