Question

For the polynomial f(x)=x^5-2x^4+8x^2-13x+6 , answer the following questions.

(a) How many zeros does the function have over the set of complex numbers?






(b) What is the maximum number of local extrema (maxima or minima) the graph of the function can have?





(c) Complete the following statements:


As , ________. As , ________.



(d) List the possible rational zeros of this function.







(e) Factor this polynomial completely over the set of complex numbers.

Answer 1

@ranga one of the last ones

Answer 2

I think I know some of these answers though.. could you check?

Answer 3

Post the answers you already have and I can check.

Answer 4

Ok... a)four zeroes b)max number=4 c)idk d)x=1,-2,1+isqrt2, 1-isqrt2 e)(x-1)^2(x+2)(x^2-2x+3)

Answer 5

How did you get four for a) ?

Answer 6

it was more sort of a guess

Answer 7

In e) they want you to factor completely and so you might as well figure out all the zeroes of the polynomial which will help answer the other parts starting with a).

Answer 8

so how do I factor e) more?

Answer 9

Use quadratic formula to find the roots of (x^2-2x+3).

Answer 10

You can't factor that with rational numbers? Would my answer be marked as wrong?

Answer 11

(e) Factor this polynomial completely over the set of complex numbers.
Since they explicitly ask you to factor over complex numbers it is necessary to do that. If they don't ask you to do that you can leave it as (x^2-2x+3).

Answer 12

Oh, but idk how to do that... what would that factor be

Answer 13

\[\frac{ -b \pm \sqrt{b ^{2} - 4ac} }{ 2a }\]
Use the above formula to find the complex roots first.

Answer 14

I can't see the formula

Answer 15

Use Firefox or Chrome browser instead of IE.

Answer 16

It is just a quadratic formula. { -b +/- sqrt(b^2 - 4ac) } / 2a

Answer 17

im sorry, could you walk me through your work? I think i'm getting confused now

Answer 18

You have already factored the polynomial into: (x-1)^2(x+2)(x^2-2x+3)
(x^2-2x+3) has complex roots. Find them using quadratic formula.

Answer 19

And from your answer to d) it looks like you have already done that and I am not sure why you are getting confused now.

Answer 20

I dont know what to write though

Answer 21

e) (x-1)^2 * (x+2) * { x - (1 + isqrt(2) } * { x - (1 - isqrt(2) }

Answer 22

As you can see from e), the polynomial has 3 real roots and 2 complex roots.
(The real root 1 has a multiplicity of 2.)

So the answer to a) is 2 because there are 2 complex roots.

Answer 23

oh ok.. are my rest right?

Answer 24

Are you in calculus? Do you know derivatives. The answer will tell me how I should answer b).

Answer 25

it should be 4 because it is less than the polynomial of ^5. Also, for e) is it supposed to be (x-(1..... )) or (x-1....)

Answer 26

So you don't want to answer my question if you know derivatives or not. So I will skip b).

If "a" is a root, then (x - a) is a factor. So if (1 + isqrt(2) is a root then { x - (1 + isqrt(2) } is a factor.

Answer 27

I'm guessing my answer for b is wrong? I think it is right, but no I do not completely understand derivatives

Answer 28

My question is: Is this a question in calculus or is this question in pre-calc or algebra question?

Answer 29

pre-calc, sorry

Answer 30

The go with 4 which is one less than the degree of the polynomial.
The same question in calculus will be approached differently.

Answer 31

Oh, but 4 is right though right?

Answer 32

In calculus you will take the first derivative, equate it to zero and solve for x.
When you take the derivative, the degree of the polynomial will go from 5 to 4. A fourth degree polynomial has a maximum of 4 roots. But when you try to solve for the zeros of the derivative and If there are only two real roots and the other two are complex then you will say the max. possible extrema is 2. So it all depends on how many real roots the derivative has. But if you have not been taught derivative yet, then all you can go by is one less than the degree of the polynomial.

Answer 33

ok thanks for explaining the calculus way. What about c)

Answer 34

c) the problem is not at all clear what is being asked.

Answer 35

For d) have you been taught the rational roots theorem?

Answer 36

for c) it is basically the x---> infinite, f(x)----> as x--->-infinite, f(x)--->

Answer 37

I know the rational roots theorem I think

Answer 38

c)
x-> infinity, f(x)-> infinity
x-> -infinity, f(x)-> -infinity

Answer 39

For d) they are asking list "possible" rational zeros of this function. They are not asking for the actual zeros but "possible rational zeros".

Use rational roots theorem. The constant is 6 and the leading coefficient is 1.
So you need to list: +/- (all factors of 6). Those are the possible roots.

Answer 40

+- 1,2,3,6?

Answer 41

d) Possible roots: 1, -1, 2, -2, 3, -3, 6 and -6.

Answer 42

right ok, thanks a lot! Hopefully, this will help me get an A later this week

Answer 43

Good luck and do well.