Question

What are the possible number of positive real, negative real, and complex zeros of
f(x) = 4x3 + x2 + 10x - 14?

Positive Real: 1
Negative Real: 0
Complex: 2
Positive Real: 1
Negative Real: 2 or 0
Complex: 2 or 0
Positive Real: 2 or 0
Negative Real: 1
Complex: 2 or 0
Positive Real: 2 or 0
Negative Real: 2 or 0
Complex: 1

Answer 1

@IMStuck or @mathmate :)

Answer 2

Do you know about the Descartes rule of signs?

Answer 3

Give me a sec and I'll help you, ok?

Answer 4

yes, ok

Answer 5

I am kind of having trouble with finding the complex ones

Answer 6

As the function stands there is only one sign change, so that means that there is only one possible positive real root. Now replace x with (-x) and see what we get for signs:

Answer 7

ok give me a sec.. :)

Answer 8

When you place a (-x) in for x your function now looks like this:\[-4x ^{3}+x ^{2}-10x-14\]and there are 2 sing changes there, so that tells us there is a possible 2 negative real roots or 0 negative real roots. Heres the table for that:

Answer 9

\[f(-x)=4(-x)^3+(-x)^2+10(-x)-14\]

Answer 10

I have a question about those signs changes. I can just count the number of times the signs changes differently right?

Answer 11

Those are your possibilities for this. It is the second set of answers you have up there.

Answer 12

Positive Real: 1 Negative Real: 2 or 0 Complex: 2 or 0

Answer 13

You can only find the complex AFTER you find the negative and positive.

Answer 14

yes that was what I was thinking. thank you so much for all this explaining. :) can you explain the complex again :/

Answer 15

I see in the table that you got the complex from the negative and positive so did you just add the numbers?

Answer 16

yes. This is a third degree polynomial so your totals have to add up to 3.

Answer 17

So here is another problem I am going to explain it to you to see if I did it right: :)

What are the possible number of positive real, negative real, and complex zeros of
f(x) = -7x4 - 12x3 + 9x2 - 17x + 3?

Positive Real: 3 or 1
Negative Real: 1
Complex: 2 or 0
Positive Real: 3 or 1
Negative Real: 2 or 0
Complex: 1
Positive Real: 1
Negative Real: 3 or 1
Complex: 2 or 0
Positive Real: 4, 2 or 0
Negative Real: 1
Complex: 0 or 1 or 3

In the positive we see that there are 3 or 1 signs changes. Then for the negative we need to plug in the -x so like this: \[-7(-x)^4-12(-x)^3+9(-x)^2-17(-x)+3 \] so it would look like: \[-7x^4+12x^3+9x^2+17x+3\] here there is 1 sign change. Then I can take the table: Therefore the answers would be the first option. And also I thank you so much cuz I will always remember you and YOU are the one that taught me all of this. So thank you so much! :)))) ;) <3 @IMStuck