Question

What are?
Rational Root Theorem,
Descartes' Rule of Signs, and
the Factor Theorem.

Answer 1

1014 = x^3 - 7x^2

Answer 2

1014 = x^3 - 7x^2

I have to solve this equation by these three methods.

Answer 3

rational roots is a way to narrow down the pool of trial and error options.

ax^2 +bx + c = 0

x^2 +bx/a + c/a = 0

the last term is important in that if this is going to come out pretty; it need to factor as c/a

Answer 4

so we take the: factors of "c"
-----------; plus and minus as our options
factors of "a"

Answer 5

Descartes' Rule of Signs is a way to figure out how many + or - zeros are possible

Answer 6

in this case, the factors of 1014
----------------
factors of 1

Answer 7

Sorry i have no idea how to do them.

Answer 8

the factors of 1 are 1 so that bottom is a bit moot

Answer 9

1014 factors into ....

Answer 10

1,1014
2,507
.. etc

Answer 11

then?

Answer 12

1 | 2 | 3 | 6 | 13 | 26 | 39 | 78 | 169 | 338 | 507 | 1014
--------------------------------------------------------------
1

Answer 13

then you have 2 options; use your pool as trial and errors in your f(x) and whatever pops out a zero is good

Answer 14

well, the other option is really the same, just a different format

Answer 15

but what's the method? i have to show! :(

Answer 16

what the f(x) ?

Answer 17

f(x) = x^3 - 7x^2 -1014

Answer 18

well, it a cubic; which tells us there are 3 zeros, some might be multiples tho

Answer 19

i know the answers, they are: 13, 13 and 6.

Answer 20

they are the sides of a cuboid.. i had to make an equation using these...
So, i used the formula of volume..

Answer 21

f(1) = 1 - 7 -1014 = not a zero

f(2) = 8 - 28 -1014 = not a zero

etcc

Answer 22

they will be 13, 13 and 6.

Answer 23

how to solve it by factor theorm?

Answer 24

f(6) = 6^3 - 7.6^2 -1014

f(6) = 216 - 7.36 -1014

f(6) = 216 - 252 -1014

f(6) = 216 - 252 -1014; you sure its a 6?

Answer 25

13 is good

Answer 26

Let me explain, i had to make an equation in terms of x.. with the volume of a cuboid.

And i have to use x+something & x + something for width and heights.

So,

Volume = L * B * H
V = 13 * 13 * 6
V = 1014

Answer 27

factor thrm i got no idea what the definition would be, but id assume its factoring it

Answer 28

Now,

Length = x
breath = x
height = (x - 7)

Answer 29

1014 = l * b* h
1014 = x * x * (x - 7)

Answer 30

that's how i made the equation.

Answer 31

ok, and x^3 - 7x^2 = 1014

x^3 - 7x^2-1014 = 0

right?

Answer 32

Yes.

Answer 33

So now?

Answer 34

so now, factor thrm is just factoring it back out to: x^2(x-7) i believe

Answer 35

but i dont get 6 then! :(

Answer 36

or rather: (x-13)(x^2+6x+78) = 0

Answer 37

Great^

Answer 38

what about another 13 and 6?

Answer 39

x = 13, not 6

Answer 40

how will will i get the height and the width?

Answer 41

you found x; use that to find the height and the width

Answer 42

width = x
length = x
height = x-6

Answer 43

since x=13 .... plug it in

Answer 44

owow^

Answer 45

So that's factor therom.. what about others? :/

Answer 46

the others is the finding a pool of options to trial and error with

Answer 47

Descartes' Rule of Signs <------------ i never heard of that. :(

Answer 48

rational roots pops out f(13) = 0 , and 13 is one of the factors of 1014

Answer 49

Ds rule of sign is simply a way to measure the max posibble number of positive or negative zeros

Answer 50

f(x) = x^3 -7x^2 -1014
----> ---------->
1 0

there is only 1 sign change which suggests a maximum of 1 positive zero



f(-x) = -x^3 -7x^2 -1014
----------------->
0

there are NO sign changes, so there are a maximum of 0 negative roots

Answer 51

by factor theorem
f(13)=0
so (x-13) is factor of x^3-7x^2-1014
now divde x^3-7x^2-1014 by x-13
qutient is x^2-6x-78
so equation becomes (x-13)(x^2-6x-78)

Answer 52

Oh! so that's ds... that was short.

why u made arrows below it?

But the sign of x^3 changed! :O

Answer 53

to show the progression from sign change to sign change and count them

Answer 54

f(-x) imples a negative value for a zero

Answer 55

f(-x) would pop out: -x^2 -7x -1014

which produces NO changes in sign thru the equation

Answer 56

i dont get it. :(

Answer 57

sorry.

Answer 58

spose g(x) = x^2 +6x +4; this would have 2 roots since its a parabola that dips under the x axis right?

g(x) = x^2+6x+4
---------->
no change in sign; max number of +roots, none

g(-x) = x^2 -6x +4
--->--->--->
1 1 0

there are 2 sign changes when x is negative; telling us there is a max possibility of 2 -roots

Answer 59

it just so happens it has 2 -roots:

http://www.wolframalpha.com/input/?i=x^2%2B6x%2B4

Answer 60

what the root values are? Ds sign rule doesnt say

Answer 61

ok, so for my problem, i should copy down this thing?

f(x) = x^3 -7x^2 -1014
----> ---------->
1 0

there is only 1 sign change which suggests a maximum of 1 positive zero



f(-x) = -x^3 -7x^2 -1014
----------------->
0

there are NO sign changes, so there are a maximum of 0 negative roots

Answer 62

If your being graded based upon material specific methods; no.

If you are being graded upon how well you understand the concepts, then yes

Answer 63

methods. :(

Answer 64

then youll have to render the appropriate material specific accounting for it. Which should be similar

Answer 65

or what do u adivce? if i make another equation and take different sizes, would that be easier?

Answer 66

it isnt the equations that are bad; its simply the method by which you are to be graded by that is the issue here. And regardless of what your equation is, the method proscribed by your material is what you are graded by.

Answer 67

without having read and studied what the author of your material does, I have no way of determining a suitable "gradable" response.

Answer 68

Alright, thanks.. if i stuck anywhere i will ask you! :)