10.a)Solve the inequality x^3-5x^2+2x+8<0 by
i) using intervals
ii) considering all cases
I did this but my teacher said that when i solved this: x^3-5x^2+2x+8<0
(x+1)(x^2-6x+8)
(x+1)(x-4)(x-2)<0
She told me to show all the steps, but i don't know how to show more. so can you please help me to show all the works and rest of the works i did too but she said " the same answer has to happen for each method. i don't really understand that. can you help me please and thank you....
assuming you could show the steps involved in factoring the polynomial, you still haven't solved the inequality.
f(x) is a cubic polynomial, so you know that
f -> -∞ as x->-∞
f -> +∞ as x->+∞
Since there are 3 roots, f changes sign between them.
So,
f(x) < 0 for x in (-∞,-1)
f(x) > 0 for x in (-1,2)
f(x) < 0 for x in (2,4)
f(x) > 0 for x in (4,∞)
You can analyze all the combinations of factors, showing how to get one factor negative, 2, or 3, and explain how f(x) is + or - depending on how many negative factors there are. But, knowing how f(x) changes sign at the roots makes thing simpler.
When using this method, though, be sure to remember that f(x) does not change sign an multiple roots of even order, because the graph just touches the x-axis there, but does not cross it.
can u please explain to me how to find these sign < >. how do u know which direction it goes to. help me please. i have a test tomorrow. so please let me know. please quick.
Sure, I can help you with that. Let's solve the inequality step by step using both interval notation and considering all cases.
i) Using Intervals:
To solve the inequality x^3 - 5x^2 + 2x + 8 < 0 using intervals, follow these steps:
Step 1: Find the critical points by setting the equation x^3 - 5x^2 + 2x + 8 = 0 and solving for x. In this case, you correctly factored the equation: (x + 1)(x - 4)(x - 2) = 0.
Step 2: Determine the sign of each factor for different intervals.
- For x < -1, all three factors are negative.
- For -1 < x < 2, only (x - 2) is positive, while (x + 1) and (x - 4) are negative.
- For x > 4, all three factors are positive.
Step 3: Create an inequality chart or number line to visualize the intervals and signs.
x < -1 -1 < x < 2 x > 4
---------------|---------|---------|---------------
Step 4: Determine if the expression is negative or positive in each interval.
- In the interval (-∞, -1), the expression (x + 1)(x - 4)(x -2) is negative.
- In the interval (-1, 2), the expression (x + 1)(x - 4)(x -2) is positive.
- In the interval (2, ∞), the expression (x + 1)(x - 4)(x -2) is negative.
Step 5: Write the solution using interval notation.
The solution is (-∞, -1) U (2, 4).
ii) Considering all Cases:
To solve the inequality x^3 - 5x^2 + 2x + 8 < 0 by considering all cases, follow these steps:
Step 1: Factor the equation (if possible). You have already correctly factored it: (x + 1)(x - 4)(x - 2) < 0.
Step 2: Identify the critical points by setting each factor equal to zero:
- (x + 1) = 0, which gives x = -1.
- (x - 4) = 0, which gives x = 4.
- (x - 2) = 0, which gives x = 2.
Step 3: Create a sign chart and test the sign of the expression for each interval created by the critical points.
x < -1 -1 < x < 2 2 < x < 4 x > 4
-------------|-------------------|-------------------|------------------|-----------
Step 4: Test a point from each interval in the original inequality to determine whether it is positive or negative.
- For x = -2, (x + 1)(x - 4)(x - 2) = (-2 + 1)(-2 - 4)(-2 - 2) = (-1)(-6)(-4) = -24, which is negative.
- For x = 0, (x + 1)(x - 4)(x - 2) = (0 + 1)(0 - 4)(0 - 2) = (1)(-4)(-2) = 8, which is positive.
- For x = 3, (x + 1)(x - 4)(x - 2) = (3 + 1)(3 - 4)(3 - 2) = (4)(-1)(1) = -4, which is negative.
- For x = 5, (x + 1)(x - 4)(x - 2) = (5 + 1)(5 - 4)(5 - 2) = (6)(1)(3) = 18, which is positive.
Step 5: Determine the combination of signs for which the expression is negative (< 0). In this case, the expression is negative when:
- (x + 1) is negative,
- (x - 4) is positive, and
- (x - 2) is negative.
Step 6: Write the solution by combining the intervals where the expression is negative.
The solution is (-∞, -1) U (2, 4).
The same answer has to happen for each method because both methods are based on the same mathematical principles and should yield the same solution. So, if you have obtained the same answer using both methods, it means that you have correctly solved the inequality.