Question

The solutions of x^3 - 6x^2 + 5x + 12 are x= -1,3, and 4.

a. Write the polynomial in x^3 - 6x^2 + 5x +12 in factored form.

b.Solve the inequality x^3 - 6x^2 + 12 <_ algebracially.

Answer 1

if the roots to some cubic equation are p, q, r

then that cubic expression factors to

(x-p)(x-q)(x-r)

Answer 2

Can you put that into the equation form for me?

Answer 3

Example:

The solutions to x^2 + 8x + 12 = 0 are x = -6 or x = -2

so x^2 + 8x + 12 factors to (x-(-6))(x-(-2)) which turns into (x+6)(x+2)

Answer 4

\[x ^{3} - 6x ^{2} + 5x + 12\]

Answer 5

Into (x+___)(x+____) ?

Answer 6

there are 3 solutions, so

(x-p)(x-q)(x-r)

(x-(-1))(x-3)(x-4) ... since p, q, and r are the 3 solutions, so p = -1, q = 3, r = 4

Answer 7

So \[(x-1)(x+3)(x+4)\] ??

Answer 8

no, the other way around actually

Answer 9

(x+1)(x-3)(x-4)

Answer 10

So that's the final answer for part a?

Answer 11

yes

Answer 12

Great thanks. Now for part b?

Answer 13

Draw a number line. Mark the points -1, 3, and 4 on this number line

Answer 14

Okay done.

Answer 15

the values on that number line that are labeled are the roots of x^3 - 6x^2 + 5x + 12

they make x^3 - 6x^2 + 5x + 12 equal to zero

Answer 16

anything else will make x^3 - 6x^2 + 5x + 12 some other number than zero

Answer 17

for example, x = 0 will make x^3 - 6x^2 + 5x + 12 equal to 12

Answer 18

so the entire region from -1 to 3 is all positive because there's no way to have it transition to negative if there are no other zeros in that region

Answer 19

what you need to do is test each region and see which ones are negative

Answer 20

So what would I test for each region?

Answer 21

From each region**

Answer 22

well x = 0 was used to test the region from -1 to 3 since 0 is between those values

Answer 23

x = -2 can be used to test this region

Answer 24

basically any number that is in that region

Answer 25

Oh okay. One sec

Answer 26

Ugh I'm stuck. So I test a number (I understand that) but what do I test it into ?

Answer 27

into f(x) = x^3 - 6x^2 + 5x + 12

Answer 28

so if x = -2, then

f(x) = x^3 - 6x^2 + 5x + 12

f(-2) = (-2)^3 - 6(-2)^2 + 5(-2) + 12

....

Answer 29

What is my outcome supposed to be?

Answer 30

what do you get when you use a calculator

Answer 31

(-2)^3 - 6(-2)^2 + 5(-2) + 12 = ???

Answer 32

I got -150.

Answer 33

its off, but at least it's negative

(-2)^3 - 6(-2)^2 + 5(-2) + 12 = -30

Answer 34

Oh jeez I was way off LOL. Oh I don't think I used parentheses.

Answer 35

Okay next step?

Answer 36

i gotcha, anyways, (-2)^3 - 6(-2)^2 + 5(-2) + 12 is negative, so x^3 - 6x^2 + 5x + 12 is negative when x = -2

so..

x^3 - 6x^2 + 5x + 12 is negative for any value less than -1