Question

Using the given zero, find one other zero of f(x). Explain the process you used to find your solution.

1 - 2i is a zero of f(x) = x4 - 2x3 + 6x2 - 2x + 5.

Answer 1

do u know what is the conjugate of a complex number, for example 1-2i?

Answer 2

nope

Answer 3

ok. if u have a complex number
\[ \large z=a+ib \]
then its conjugate is
\[ \large \overline{z}=a-ib \]
u have a complex root z=1-2i. what would its conjugate be?

Answer 4

i have no clue :(

Answer 5

its conjugate would be 1+2i
u just have to change the sign of whatever is with the \(i\)

Answer 6

ohh ok so first change the sign way after

Answer 7

now. when a polynomial has a complex root (such as 1-2i in your case) then its conjugate is also a root of the polynomial (1+2i in this case).

Answer 8

this means that your polynomial can factor into
\[ \large f(x)=(x-(1-2i))(x-(1+2i))g(x) \]
ok?

Answer 9

okk

Answer 10

u just have to find g(x) in whichever way u know

Answer 11

which is the best way to do it the easiest?

Answer 12

it depends on u: either long division or synthetic division

Answer 13

whatever u find easiest

Answer 14

can show me how to do synthetic

Answer 15

it's gonna take long to do it with the keyboard

Answer 16

give me a second

Answer 17

ok then long division if its easier

Answer 18

what ever works for u

Answer 19

i m gonna show u synthetic division
just give a minute to type everything correctly

Answer 20

ok would it be esier to draw

Answer 21

no way

Answer 22

\[ \large \begin{array}{c|ccccc}
& 1 & -2 & 6 & -2 & 5\\
1-2i & & 1-2i & -5 & 1-2i & -5\\
\hline
& 1 & -1-2i & 1 & -1-2i & 0\\
1+2i & & 1+2i & 0 & 1+2i\\
\hline
& 1 & 0 & 1 & 0
\end{array} \]

Answer 23

this means that
\[ \large g(x)=x^2+1 \]

Answer 24

got it?

Answer 25

thats all?

Answer 26

what r the roots (zeros) of g(x)?
that's what u r asked for

Answer 27

2

Answer 28

no

Answer 29

\[ \large 0=g(x)=x^2+1 \]
then
\[ \large x^2=-1 \]

Answer 30

so -1

Answer 31

or -1/2

Answer 32

no :(
\[ \large x=\pm\sqrt{-1}=\pm i \]

Answer 33

just i?

Answer 34

+i and -i
it's a quadratic equation

Answer 35

ohh so

Answer 36

\[ \large \pm\ \sqrt{-1} \]

Answer 37

u have two solutions

Answer 38

i have no clue wats the other 1

Answer 39

wait i thought i only find 1 of the other zero

Answer 40

the four zeros (roots) of the polynomial are
\[ \large 1-2i\qquad\text{given} \]
\[ \large 1+2i \]
\[ \large i \]
\[ \large -i \]

Answer 41

right

Answer 42

so u r done!!

Answer 43

ohhhh

Answer 44

did u understand everything?

Answer 45

ya thanks to you ur the BEST

Answer 46

u r welcome