Question

I need help with Rational Root Theorem and Descartes Rule of Signs. What exactly is the Rational Root Theorem and Descartes Rule of Signs? and how do I answer these three questions??
1. What are the possible rational zeros of f(x) = x^4 + 6x^3 - 3x^2 + 17x - 15?
2. What are the possible rational zeros of f(x) = 3x^4 + x^3 - 13x^2 - 2x + 9?
4. What are the possible number of positive real, negative real, and complex zeros of
f(x) = 4x3 + x2 + 10x - 14?

Answer 1

@phi
Can you walk me through how to do these problems please?

Answer 2

Here is Descartes
http://www.purplemath.com/modules/drofsign.htm

Answer 3

okay I have a better understanding of Descartes

Answer 4

For questions 1 and 2, you use the rational root theorem
This video goes into it.
http://www.youtube.com/watch?v=YMyv9-9VXw4

Answer 5

Thanks be right back, going to watch it.

Answer 6

@phi
After watching I solved #1 and came of with the numbers 1,6,-3,17,-15 as possible rational zeros.

Answer 7

I think I did it correctly...
1. What are the possible rational zeros of f(x) = x^4 + 6x^3 - 3x^2 + 17x - 15?

± 1, ± 3, ± 5, ± 15

± 1, ± 1 over 3, ± 1 over 5, ± 1 over 15

± 1, ± 3, ± 6, ± 15, ± 17 <-- is the answer I would choose.

± 1, ± 1 over 3, ± 1 over 6, ± 1 over 15, ± 1 over 17

Answer 8

The Rule is "factors of the constant term" (the number with no x next to it)
divided by the "factors of the highest-order term" (factor of the x to the biggest exponent)

in this case you look at 1x^4 and -15, and see you want 15 and 1

it will be the factors of 15 divided by the factors of 1.
factors of 15 are
1,3,5,15
they can be ± , so we could list them as
-1,-3,-5,-15,1,3,5,15

Answer 9

Notice you do not use *all* the coefficients (which is what you did). You pick that very last number, and the very first coefficient (if the polynomial is written in standard form)

Answer 10

For Question 2)
f(x) = 3x^4 + x^3 - 13x^2 - 2x + 9

what two numbers do you use?

Answer 11

3 and 9?

Answer 12

yes. which goes "up top" ?

Answer 13

1 and 3

Answer 14

the factors of 9 (the last number in 3x^4 + x^3 - 13x^2 - 2x + 9)
goes up top
the factors of 3 (the first coefficient, from 3x^4 ) goes in the bottom
\[ \frac{\text{ factors of 9}}{\text{factors of 3} }\]

Answer 15

with ±

Answer 16

yes \[\pm 1 \pm 3\]
is that correct or would \[\pm 9\] be included

Answer 17

the factors of 9 are
1,3,9 (the 3*3 =9 repeats the 3. we need to list it only once)

the factors of 3 are
1,3

now find all combinations of
1,3, 9 divided by 1 and then by 3

Answer 18

in other words, you want
1,3,9 divided by 1, which gives 1,3,9
then you want
1,3,9 divided by 3, which gives 1/3, 1,3
merge the list, and ignore duplicates

what do you get ?

Answer 19

± 1, ± 1/3, ± 3, ± 9

Answer 20

that is the answer. Can you go over the process, and see how you get it ?

Answer 21

Yes, Sir I have to keep really good notes cause my work is online and its harder to learn math. So to learn I've got to keep well written steps on how to find things.

Answer 22

I don't see Question 3.
Question 4 uses Descartes rule of signs.

Answer 23

Question 3 was graphing a function. and I already knew how to do that...

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

A.
Positive Real: 1
Negative Real: 0
Complex: 2

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

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

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

Answer 24

@phi

Answer 25

See if this helps
http://www.youtube.com/watch?v=MVX2vYlc24k

Answer 26

ok