Question

For the function g(x)=x^5-16x^3 solve the following. g(x)>0

Answer 1

Can you factor anything out from each term?

Answer 2

trickets u never responded to mkuehn. Look at what is similar in each terms.

Answer 3

x can be factored out?

Answer 4

not just x, x to some power.

Answer 5

If you can factor out x, try factoring out x^2. Can you factor out even more than x^2 ?

Answer 6

factor out x^3? which gives me x^2-16?

Answer 7

Well it gives you this:

(x^3)(x^2 - 16) > 0

Answer 8

So now we basically want to find the zeros of this function. Can you factor anything else in order to figure out the zeros of this function?

Answer 9

true so now factor the other side (hint a^2-b^2 = (a+b)(a-b) )

Answer 10

sorry guys, im just not getting this... so my equation now is (x^3)(x^2-16)>0 ?

Answer 11

Yes can you factor the x^2 - 16 ?

Answer 12

If you can imagine the graph of this function, it is a cubic, so it goes up and down and up and down along the x-axis, we are trying to figure out all the places where it crosses the x-axis and then we can determine which intervals it is positive.

Answer 13

rewrite the x^2-16 as x^2-4^2
Then use: a^2-b^2=(a-b)(a+b)

Answer 14

where a=x and b=4

Answer 15

okay i think mkuehn is doing a good job, i will let him finish

Answer 16

so (x-4) and (x+4)?

Answer 17

Yes! Exactly!

Answer 18

So now to find the zeros we have (x^3)(x-4)(x+4) Set each of those factors equal to zero and solve for x.

Answer 19

x^3 = 0 x = ?

x + 4 = 0 x = ?

x - 4 = 0 x = ?

Answer 20

x=0? on all of them?

Answer 21

wait no 0, -4, 4?

Answer 22

It works for the first one.

If x = 0 for the second one then 0 + 4 = 0 ? Does that make sense? What value of x + 4 equals 0?

Answer 23

You are answering the question what number plus four equals zero ?

Answer 24

0, -4, and 4?

Answer 25

Yes!

Answer 26

So now we know this is where our graph crosses the x-axis, so we basically have 4 intervals we need to look at to see if the function is positive or negative.

Answer 27

From negative infinity to -4

From -4 to 0

From 0 to 4

From 4 to infinity

Answer 28

ok im starting to follow along..

Answer 29

So what we can do here is pick a value inside of each interval and evaluate the function. If the value we get back is positive then we know g(x) is greater than 0 inside that interval. If we get a value that is negative then we know g(x) is less than 0 inside that interval.

Answer 30

so (-oo,-4)U(4,oo)?

Answer 31

I'll show you the first example. For the interval from negative infinity to -4 I will select -5.

g(-5) = (-5)^5 - 16(-5)^3 = -3125 - 16(-125) = -3125 + 2000 = -1125

So from negative infinity to -4 the function is less than 0. We don't want that, so that is not part of the solution.

Answer 32

Now check a value between -4 and 0 to check to see if it is positive or negative.

You are on the right track.

Answer 33

Great job mkuehn10!

Answer 34

u too tricketts8737

Answer 35

positive?

Answer 36

Check out this picture of the function for reference.

YES IT IS POSITIVE, so the interval from -4 to 0 is part of the solution.

Answer 37

Hmm that graph doesn't look right.

Answer 38

you have -x^5

Answer 39

Thank you.

Answer 40

There, that should make it a lot clearer.

Answer 41

(-oo,-4)U(0,4)?

Answer 42

So as we figured out it is negative from negative infinity to -4 and positive from -4 to 0.

We just need to figure out from 0 to 4 and 4 to infinity. Can you tell by looking at the graph?

Answer 43

That interval is where it is less than zero.

Answer 44

We want greater than zero. You are right on the cusp of getting it.

Answer 45

(-4, 0) U (0,oo)

Answer 46

You can see this on the graph I attached as well as proving it by checking values inside the intervals.

Answer 47

ahhh... ok a visual graph would help me alot, i just have a hard time understanding this!

Answer 48

Why are the zeros not included in the intervals?

Answer 49

Did you see the file I attached? I will attach it to this message too.

Answer 50

yes! i see it

Answer 51

Pretty easy to see where it is negative and positive there. And we showed it with the math.

Answer 52

so that makes it (-oo,-4)U(4,oo)?

Answer 53

If you go from left to right on the graph you see it is negative from -infinity to -4 then it is positive from -4 to 0 negative again from 0 to 4 and positive from 4 to infinity

Answer 54

Since we want where the function is greater than zero, it corresponds with the intervals where the function is positive.

From -4 to 0 and from 4 to infinty

(-4, 0) U (4, oo)

Answer 55

ohh ok that makes since.. now that i put the visual concept behind it!! thank you so much!

Answer 56

You're welcome!