10.a)Solve the inequality x^3-5x^2+2x+8<0 by

Question

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....

Answer 1

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.

Answer 2

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.