Question

attach file,I need help number 4,18,20,22,TY

Answer 1

18. one real root
20 two real roots

Answer 2

best way it so graph and see where the roots are

Answer 3

but teacher require find root by negative and positive sign

Answer 4

I'll do problem 20. Hopefully it makes sense!
\[q(x)= x^{4}+5x^{3}+2x^{2}-7x-9\]
First, note the highest exponent=4, so there is a total of 4 zeros.
Let's use Descarte's rule of signs for positive roots.
signs: +++-- (ignore everything except the signs. Be sure the terms are in order, highest order to lowest order)
Count the sign changes. One way is to write the signs in pairs
(++), (++),(+-),(--)
Only 1 pair has a +-, so only 1 sign change. ==> 1 positive root.

Negative roots: sub in -1 for x, and count sign changes
signs: +-++-
in pairs: (+-),(-+),(++),(+-)
3 pairs have sign changes, ===> 3 negative roots, or 1 negative root

Now, using the fact that we have a total of 4 roots, we can list out the possibilities. Remember, imaginary roots come in pairs.

pos neg imag
1 3 0
1 1 2

As a check, use wolfram
http://www.wolframalpha.com/input/?i=roots+x%5E4%2B5x%5E3%2B2x%5E2-7x-9

We see the actual answer is 1 positive, 1 negative, 2 imaginary.

Answer 5

Phi if the pair opposite site it mean poistive root ?

Answer 6

you mean opposite signs?
If a pair has opposite signs it means there was a change of signs in the sequence. It's a way to count the number of sign changes.

Answer 7

You can count sign changes another way.
++++----- has only one change where + changes to -
Another example
+-+ has 2 changes + to - and then - to +

Answer 8

you have time , we can work another number?

Answer 9

Go ahead

Answer 10

let try number 22)

Answer 11

positive: (+ - ) (-+) (+-)(-+)(+-)
negative; (+-) (-+) (+-)(-+)(+-)
than what root positive have ?
Negative what root have ?

Answer 12

Good so far. How many sign changes for pos and neg?

Answer 13

dose or dose'n matter in oder like
postive pair positve go first : (+-)
negaitve pair negative gof first : ( -+)?

Answer 14

Order does not matter.

We have 5 sign changes for the positive roots. This means the equation has either 5 positive roots, or 3 positive roots, or 1 positive root.

Answer 15

look like not change from positive to negative

Answer 16

OK. Just look at the positive for now. How many sign changes for the positive. That means how many pair have (+ -) or (-+)?

Answer 17

positive has 5 sign change

Answer 18

So to repeat, that means we have 3 possibilities: 5, 3, or 1 positive root. We do not know which, only that these are the possible choices.

Answer 19

Now count the sign changes for the negative, and list out the possible number of negative roots.

Answer 20

5 sign change
5 id odd ,than we use all odd number ? like
5,3,1, ?

Answer 21

Good.

We go down by 2 for our choices, because the other roots are complex, and they come in pairs.

Answer 22

Now we make a table of all possible combinations. We know we have 10 roots because the highest order term is x^10.

Answer 23

I'll list the first possibility

Pos Neg Complex
5 5 0

Answer 24

if 3
Pos Neg Complex
3 ? ?

Answer 25

To make I sure I get every combination, I hold the first number (Positive) constant, and then use up all the combinations for the negatives.
So I would next list
Pos Neg Complex
5 3 ?
and then
5 1 ?

I left ? for the complex because you can figure them out.
We are not done yet...

Answer 26

Pos Neg Complex
5 3 2
and then
5 1 4
?

Answer 27

OK, now select the next possibility for the positive, and then all possibilities for the negative.

Answer 28

Pos Neg Complex
3 3 4
and then
3 1 6

Answer 29

you forgot 3 5 2
but keep going...

Answer 30

Pos Neg Complex
3 3 4
3 5 2
and then
3 1 6

Answer 31

one more possibility for the positive...
3 more combinations to go.

Answer 32

Pos Neg Complex
1 3 6
1 5 4
1 6 3
1 4 5

Answer 33

Hmmmm...

Answer 34

The possibilities for the number of negative roots are 5, 3 or 1.

Answer 35

So some on your list are not possible.

Answer 36

We have those combinations already. Set the pos to 1, and use all the neg (5,3,1)

Answer 37

Pos Neg Complex
1 3 6
1 5 4
1 1 8

Answer 38

So let's list all the possibilities. I'll list them the way I would do it:
Pos Neg Complex
5 5 0
5 3 2
5 1 4
3 5 2
3 3 4
3 1 6
1 5 4
1 3 6
1 1 8

Answer 39

Now the actual number of roots can be found using wolfram
http://www.wolframalpha.com/input/?i=roots+x%5E10+-+x%5E8+%2Bx%5E6+-+x%5E4+%2Bx%5E2-1

Answer 40

ty,can you work through number 18 ?

Answer 41

Work on it, and I'll check you. Post the intermediate results, to make sure you have not drifted off course.

Answer 42

g(x) =(++)(+-)(-+)
g(-x)=(-+)(++))(++)
positive sign has 2 roots
negative sign has 1 root
Pos Neg Complex
2 1 0
1 2 0

Answer 43

You counted the number of sign changes correctly

Answer 44

for the positive, you found 2, so the choices are 2 and 0

Answer 45

yes , I can count number sign chage ,but I don't know how many root are possible

Answer 46

The choices are always the number of sign changes = n, and then n-2, n-4, until you get to zero.

Answer 47

Try again with the table.

Answer 48

Pos Neg Complex
2 1 0
2 0 1
0 2 1
0 0 2

Answer 49

OK, we have to work on this.

Answer 50

my answer is wrong ?

Answer 51

The choices for the number of positive roots are {2, 0}
The choices of the number of negative roots are {1}.
When you make your table, no other numbers are allowed. For the moment, ignore the complex. just list the combinations for the pos and neg.

Answer 52

Pos Neg Complex
2 1 0
2 0 1
0 2 1
0 1 2

Answer 53

In your list you have 0 and 2 in the neg column, but neither is in {1}, which is the number of sign changes we counted. So still not correct..

Answer 54

Pos Neg Complex
2 1 0
0 1 2

Answer 55

Yes!

Answer 56

than 2 possible ?

Answer 57

The 2 in the positive column is ok because you counted 2 sign changes.
The 2 in the complex column is just the number so the total comes out right.

Answer 58

I will post a problem fog, if you still around Please! help

Answer 59

fog?

Answer 60

yes, I can draw but I don't know posissble out come, fog,gof ,how many oder pair

Answer 61

OK post your problem

Answer 62

OK, I will open file and post now