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....

Steve explained this, but still i didn't got the steps. can someone do it for me step by step so i can understand the steps of it please.

Ok, so all the algebra work has been done

x^3-5x^2+2x+8<0
(x+1)(x-4)(x-2)<0

Quickest way:
I would look at the graph of
y = x^3-5x^2+2x+8
= (x+1)(x-4)(x-2)
I know that this is standard cubic function opening up in quadrant I and having x-intercepts of -1, 2, and 4
So a very rough sketch shows me that this curve is below the x-axis ( < 0 ) for x < - 1 or 2 < x < 4

looking at all cases , the critical values of -2,2, and 4 split the line into 4 parts :

if x < -1, try some value, e.g. x = -5
(-)(-)(+) < 0 , so yes
x < -1

if x is between -2 and 2 , e.g. x = 0
(+)(-)(-2) > 0 , so NO

if x is between 2 and 4 , e.g. x = 3
(+)(-)(+) < 0 , so YES
x is between 2 and 4

if x > 4 , e.g. x = 5
(+)(+)(+) > 0 , NO

then x < -2 OR 2 < x < 4

Reiny help me with using intervals. u forgot to help me with that. its urgent please

Sure! I'll break down the solution step by step for you.

i) Solving the inequality by using intervals:

Step 1: Factorizing the polynomial expression: x^3 - 5x^2 + 2x + 8
To factorize it, you correctly identified that (x + 1), (x - 4), and (x - 2) are the factors. Good job!

So, the factored form is (x + 1)(x - 4)(x - 2) < 0.

Step 2: Determining the critical points:
Critical points are the values of x where the expression changes its sign. In this case, the critical points are -1, 2, and 4.

Step 3: Creating intervals and checking the sign:
We will create intervals using the critical points and test the sign of the expression within each interval.

Interval 1: (-∞, -1)
Choose a test value less than -1, let's say x = -2:
Substitute x = -2 into the factored expression: (-2 + 1)(-2 - 4)(-2 - 2) = (-1)(-6)(-4) > 0

Since the expression is positive in Interval 1, it does not satisfy the inequality.

Interval 2: (-1, 2)
Choose a test value between -1 and 2, let's say x = 0:
Substitute x = 0 into the factored expression: (0 + 1)(0 - 4)(0 - 2) = (1)(-4)(-2) < 0

Since the expression is negative in Interval 2, it satisfies the inequality.

Interval 3: (2, 4)
Choose a test value between 2 and 4, let's say x = 3:
Substitute x = 3 into the factored expression: (3 + 1)(3 - 4)(3 - 2) = (4)(-1)(1) < 0

Again, the expression is negative in Interval 3, so it satisfies the inequality.

Interval 4: (4, ∞)
Choose a test value greater than 4, let's say x = 5:
Substitute x = 5 into the factored expression: (5 + 1)(5 - 4)(5 - 2) = (6)(1)(3) > 0

Since the expression is positive in Interval 4, it does not satisfy the inequality.

Step 4: Final solution:
From the intervals analysis, we found that the inequality (x + 1)(x - 4)(x - 2) < 0 is satisfied in Interval 2 and Interval 3.

Therefore, the solution to the inequality is: -1 < x < 2 and 4 < x.

ii) Solving the inequality by considering all cases:

To satisfy the inequality (x + 1)(x - 4)(x - 2) < 0, we need to consider all possible combinations of the factors being positive or negative.

Case 1: (x + 1) < 0, (x - 4) > 0, (x - 2) < 0
In this case, the solution is: -1 < x < 2

Case 2: (x + 1) < 0, (x - 4) < 0, (x - 2) > 0
In this case, there is no solution because the product of three negative factors is positive.

Case 3: (x + 1) > 0, (x - 4) > 0, (x - 2) < 0
In this case, there is no solution because the product of two positive factors and one negative factor is negative.

Case 4: (x + 1) > 0, (x - 4) < 0, (x - 2) > 0
In this case, the solution is: x > 4

Combining all cases, we have two intervals of solutions: -1 < x < 2 and x > 4.

That's it! You have successfully solved the inequality using both interval analysis and considering all possible cases. Remember that it's important to understand the steps involved and how each method provides the same solution.