Question

Solve using intervals. x^3+6x^2+7x+12 >0

Answer 1

You will need to find the Real Zeros of the cubic expression. There are not that are Positive. Do you know why that is easy to see?

Answer 2

Yes. I tried using factors of 12, negative ones, but I was not getting 0

Answer 3

Well, isn't that fun! Okay, you proved that there are no RATIONAL zeros. Now, the task is to find one AT ALL. The HAS to be one REAL zero. In your Rational Research you should have gotten:

x = -4 ==> Something Positive
x = -6 ==> Something Negative

True? If not, do those two and show your results.

Answer 4

16 and -30

Answer 5

What does this tell you about the location of a REAL zero?

Answer 6

uhhh Im not sure..

Answer 7

If you graph y = x^3+6x^2+7x+12, can you get from (-6,-30) to (-4,+16) without crossing the x-axis?

Answer 8

I believe not

Answer 9

This is important. It means that there MUST be a REAL zero between x = -6 and x = -4.

Note, because we already know this is the ONLY Real Zero, we can already state much fo the answer.

The expression is less than zero for x <= -6
The expression is greater than zero for x >= -4
We just don't quite know enough about -6 < x < -4

Try x = -5 and see if it tells us anything. We know it won't produce zero, but maybe it will be close.

Answer 10

gave me 2

Answer 11

Barely positive. Now we know:

The expression is less than zero for x <= -6
The expression is greater than zero for x >= -5
We just don't quite know enough about -6 < x < -5

Can you see how to narrow it down some more?

If it were me, I might be tempted to try 2/32 = 1/16 = 0.0625 and give -5.0625 a try. (That was called "Linear Interpolation". Have you met it?)

Answer 12

aha the answer is -5.06, you were right. But no I don't know what linear interpolation is

Answer 13

How did you get 2/32? :S How did you know you had to do that

Answer 14

It is just an educated guess. Given (-6,-30) and (-5,2) we are 32 apart. That's 30 on the negative side and 2 on the positive side. You can set up a proportion.

(-5,2) -- We know this.
(W,0) -- We want to know this.
(-6,30) -- We know this.

Then, the educated guess is represented by this: \(\dfrac{-5-W}{-5-(-6)} = \dfrac{2-0}{2-(-30)}\)/

After some simplification, we have: \(-5-W = \dfrac{2}{32}\)

This is easily solved: \(W = -5 - \dfrac{2}{32} = -5 - \dfrac{1}{16} = -5.0625\)

Pretty sneaky, no?

Actually, the answer is -5.08772982227585. It would be a great miracle, indeed, if linear interpolation took us directly to the exact value. With a little luck, it should get us closer.

Answer 15

aha wow that's amazing ! Thank you so much :) I'm going to print that and keep it in my book

Answer 16

Good call. There are many, MANY even fancier methods for this sort of thing. Keep "Linear Interpolation" in your bag of tricks. It's a good start to many interesting explorations.

Answer 17

Alright I"ll remember it ! Thanks so much ! :)