Question

Use this polynomial, f(x)=3x^4-11x^3+10x-4
To find all possible rational zeros, to describe the roots of the polynomial, find an upper bound, find a lower bound, state all the solutions over the set of complex numbers, factor the polynomial completely over the set pf complex numbers, and prove the linear factors produce the polynomial.

If I have someone just to guided me would be Helpful. WILL FAN AND MEDAL.

Answer 1

I know its a lot but I do need help.

Answer 2

i can help with the first part

Answer 3

Okay :)

Answer 4

all possible rational zeros, numbers that divide 4 (the constant)

Answer 5

oops i didn't read it carefully scratch that

Answer 6

all fractions where the numerator divides 4 (the constant) and the denominator divides 3 (the leading coefficient)

Answer 7

the list is kind of long \[\pm1,\pm2,\pm4\] and then all of those over \(3\)

Answer 8

You use those numbers because those are factrs of +/- 4

Answer 9

over 3 cause 3 is leading coefficiannt

Answer 10

yes

Answer 11

yes

Answer 12

so also include\[\pm\frac{1}{3},\pm\frac{2}{3},\pm\frac{4}{3}\] that is all

Answer 13

Oh duh.. I knew that.. So, those would be rational zeros,?

Answer 14

OKAY. Thanks, can you explain Decartes Rule by any chance?

Answer 15

oh no
it is a cubic polynomial, it has at most 3 real zeros
that is a list of POSSIBLE RATIONAL zeros

Answer 16

ok i mis read it again
it is a polynomial of degree 4
it has at most 4 real zeros

Answer 17

yes we can look at descartes rule of sine

Answer 18

I knew it had at most 4 cause its quadtratic in its exponent :) And sine or sign?

Answer 19

it is not a "quadratic"
i know quad is often used to mean "4" like in "quadrophonic" but a quadratic has degree 2
this has degree 4 don't say it is quadratic

Answer 20

descartes rule of sign

Answer 21

Oh. Yeah, Its a weird name but yes. THANKS for correcting me.

Answer 22

\[ f(x)=3x^4-11x^3+10x-4\] has as 3 changes in sign of the coefficients

Answer 23

Because of the '-' and '+' changing 3 times?

Answer 24

from \(+3\) to \(-11\) is one change
from \(-11\) to \(+10\) is another
from \(+10\) to \(-4\) is the third

Answer 25

Gotcha :)

Answer 26

that tells you it has 3 positive zeros or 1 (you count down by twos, and since it cannot have \(-1\) positive zeros it either has 3 or 1

Answer 27

i.e. it MUST have one positive zero, maybe 3

Answer 28

now for the negative zeros, look at the number of changes in sign of \(f(-x\)

Answer 29

\[f(x)=3x^4-11x^3+10x-4\]
\[f(-x)=3(-x)^4-11(-x)^3+10(-x)-4\]\[=3x^4+11x^3-10x-4\]

Answer 30

how many changes in sign do you see ?

Answer 31

+3 + +11 is first
+11 - 10 is two
-10 - 4 is third, right?

Answer 32

no

Answer 33

or just 2?

Answer 34

CHANGES in sign
from \(+\) to \(-\) or \(-\) to \(+\)

Answer 35

yeah it only does that twice at the beginning.

Answer 36

i mean once.. right?

Answer 37

Or am i over thinking now

Answer 38

look carefully
yes
once

Answer 39

okay now I see

Answer 40

from \(+11\) to \(-3\)

Answer 41

ts wanting to see changes in addition to subtraction or vice versa

Answer 42

ok not really
from \(+11\) to \(-10\)

Answer 43

and in how many times, to tell us the roots we may have

Answer 44

for \(f(-x)\) it counts the NEGATIVE roots

Answer 45

So -11+10 only counts?

Answer 46

Ahh. and -11 is the first negative

Answer 47

So we start from there

Answer 48

are you looking at \(f(x)\) or at \(f(-x)\)?

Answer 49

Oh I was looking at the wrong one, I am sorry. Scrolled up too far. I am so sorry

Answer 50

So its looking for the first subtraction part?

Answer 51

\[f(x)=3x^4-11x^3+10x-4\] three changes
\[f(-x)=3x^4+11x^3-10x-4\] one change

Answer 52

For negative roots?

Answer 53

one change for \(f(-x)\) means one negative root

Answer 54

three changes for \(f(x)\) means three or one positive roots

Answer 55

Okay, I see. I had to process that. Now, is it always true for f(-x) to have one negative root, or can there also be 3 negative rots, depending on the problem?

Answer 56

there could be as many as there are changes in sign

Answer 57

That one just happened to have 1.

Answer 58

for negative, anyways

Answer 59

and you count down by twos
so for example if there were 5 changes in sign of \(f(-x)\) there could be 5,3 or 1 negative root

Answer 60

same with positive roots when looking at the changes of sign of \(f(x)\)

Answer 61

I see, Thats why there is 3 or 1 positive for the f(x) positive

Answer 62

in your example only one change in sign of \(f(-x)\) so one negative root

Answer 63

I see now. That rule isnt too bad!

Answer 64

got stuck sorry

Answer 65

now to find the zeros i guess you have to factor this

Answer 66

by some miracle it factors as \[(3 x-2) (x+1) (x^2-4 x+2)\] so the zeros are easy to find

Answer 67

@satellite73 would that be considered the solutions?

Answer 68

Whaever makes those =0