Question

State how many imaginary and real zeros the function has.

f(x) = x3 - 20x2 + 123x - 216

0 imaginary; 3 real

2 imaginary; 1 real

3 imaginary; 0 real

1 imaginary; 2 real

Answer 1

I got 3 sign changes

Answer 2

sign changes relate to POSSIBLE zeros ... actual zeros are trickier

Answer 3

like how

Answer 4

ur gonna have to find atleast 1 real zero and do synthetic division

Answer 5

look at the first and last coefficients
1 and 216

Answer 6

x3 - 20x2 + 123x - 216
^ ^ ^
3 sign changes, so 3 poosible reals or 1 possible real and 2 imaginaries


opposite sign the odd powers

-x3 - 20x2 - 123x - 216

0 sign changes suggests that there are 0 possible negative roots

Answer 7

try +/- 1 2 3

Answer 8

3 works

Answer 9

so it is 0 imaginary 3 real

Answer 10

its either a 0:3 or a 2:1 ratio

Answer 11

right so its 0:3

Answer 12

it does turn out that way, but how do we prove it?

Answer 13

by counting sign changes of both

Answer 14

the conjugate an the original