solve the following inequality. write the answer in interval notation
x^3-4x^2-21x<0
please show work so i can understand this problem better! thanks
Since x^3-4x^2-21x = x(x+3)(x-7)
there are no repeated roots. So, at each root, the graph crosses the x-axis, and the function changes sign.
Starting with large negative x, all 3 factors are negative. So, on the number line, start with - signs and switch at each root:
----- -7 +++++ 0 ----- 3 +++++++
x is in (-oo,-7) U (0,3)
To solve the inequality x^3 - 4x^2 - 21x < 0, we need to find the intervals where the expression is less than zero. Here's the step-by-step process to solve it:
Step 1: Factor the inequality if possible.
Unfortunately, the given inequality cannot be easily factored. Therefore, we'll have to try a different approach.
Step 2: Find the critical points.
To find the critical points, set the expression x^3 - 4x^2 - 21x equal to zero and solve for x:
x^3 - 4x^2 - 21x = 0
Here, we can factor out an x:
x(x^2 - 4x - 21) = 0
Now, we have two factors: x = 0 and x^2 - 4x - 21 = 0.
To find the solutions to the quadratic equation x^2 - 4x - 21 = 0, we can factor it as well:
(x - 7)(x + 3) = 0
So, the critical points are x = 0, x = 7, and x = -3.
Step 3: Determine the sign of the expression in each interval.
To determine the sign of the expression x^3 - 4x^2 - 21x, we test a value from each interval into the inequality.
Interval 1: x < -3
If we choose x = -4, we can substitute it into the expression:
(-4)^3 - 4(-4)^2 - 21(-4) = -48 - 64 + 84 = -28
Since the result is negative, the expression is negative for all x-values less than -3.
Interval 2: -3 < x < 0
If we choose x = -2, we can substitute it into the expression:
(-2)^3 - 4(-2)^2 - 21(-2) = -8 - 16 + 42 = 18
Since the result is positive, the expression is positive for all x-values between -3 and 0.
Interval 3: 0 < x < 7
If we choose x = 4, we can substitute it into the expression:
4^3 - 4(4)^2 - 21(4) = 64 - 64 - 84 = -84
Since the result is negative, the expression is negative for all x-values between 0 and 7.
Interval 4: x > 7
If we choose x = 8, we can substitute it into the expression:
8^3 - 4(8)^2 - 21(8) = 512 - 256 - 168 = 88
Since the result is positive, the expression is positive for all x-values greater than 7.
Step 4: Write the answer in interval notation.
Based on the signs of the expression in each interval, we can conclude that the solution to the inequality x^3 - 4x^2 - 21x < 0 is:
(-∞, -3) U (0, 7)