Question

Use the Rational Zeros Theorem to write a list of all possible rational zeros of the function.

\(f(x) = 3x^3 + 39x^2 + 39x + 27\)

Answer 1

what does the rational root thrm tell us?

Answer 2

This is what my lesson says but I don't really get what it means.
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Answer 3

fair enough, in short, take the coefficent of the highest degree term, and the constant term, and use them to produce a pool of options

Answer 4

so, do you agree that 3 and 27 are the values we need to use?

Answer 5

yes

Answer 6

then when we factor them and set them up as a rational number:

factors of 3
----------
factors of 27

the different combinations result in a pool of the only possible rational zeros that it can have

Answer 7

3 - 1,3
27 - 1, 3, 9, 27

Answer 8

so\[\pm\frac{1,3}{1,3,9,27}\]
ignoring signs, there are at best 4(2) options for possible roots

Answer 9

in proper, 1/1 and 3/3 are equal so we wouldnt want to list them both, and just use the simplest forms

Answer 10

So is that all they want or is there more?

Answer 11

they want it formated into a proper list, but yeah, thats the brunt of it all

Answer 12

do you understand how to find the elements of the list?

Answer 13

1/1, 1/3, 1/9, 1/27, 3/1, 3/3, 3/9, 3/27?

Answer 14

yes, but its not 'proper' yet.

Answer 15

reduce and just list duplicate elements once

Answer 16

and include each term as postive and negative

Answer 17

ok and I need to use \[\pm \] in front of each one right?

Answer 18

i would, yes

Answer 19

\(\pm1, \pm1/3, \pm1/9, \pm1/27, \pm3,\)

Answer 20

let me verify

1/1, 1/3, 1/9, 1/27, 3/1, 3/3, 3/9, 3/27

1, 1/3, 1/9, 1/27, 3, 1, 1/3, 1/9

1, 1/3, 1/9, 1/27, 3

and to be proper we might need them sorted

1, 1/27, 1/9, 1/3, 3

Answer 21

Thank you so much for all of your help!

Answer 22

good luck

Answer 23

I think there should be 6
@amistre64 do you think you missed one or do you think I have it wrong
using the Descartes rule for signs
f(x) + + + + no change in sign 3 roots none positive real
f(-x) - + - + 3 changes in sign 3 real negative roots