Question

Create a polynomial function of at least degree 3, with rational coefficients and roots of:

Answer 1

\[x1=\frac{ 3-\sqrt{2} }{ 2 }\]\[x2=\frac{3+\sqrt{2} }{ 2 }\]\[x3=2i\]

Answer 2

@DanJS @Nnesha @Directrix

Answer 3

i have 3 minutes O_O

Answer 4

but you can take five

Answer 5

@Zale101

Answer 6

do you have to do this today

Answer 7

tag some people
i gtg sorry
and i know its gonna take at least 10 mintues

Answer 8

Just create 3 terms out of the roots and multiply it out

Answer 9

How do I do that?

Answer 10

@jim_thompson5910

Answer 11

@Nnesha
got time now?

Answer 12

still there dtan?

Answer 13

we will do this today

Answer 14

I can go thru this now

Answer 15

go ahead

Answer 16

@alekos

Answer 17

OK lets call the roots a,b and c

Answer 18

alright

Answer 19

so we have three terms (x-a), (x-b) and (x-c) which when we multiply together we will get our cubic polynomial

Answer 20

(x-a)(x-b)(x-c) = dx^3 + ex^2 + fx + g

Answer 21

so i do multiply them all out in fraction form?

Answer 22

we just need to find the rational coefficents d,e,f and g

Answer 23

so what should a and b be?

Answer 24

lets start with (x-a)(x-b) where a = (3-sqrt2)/2 and b = (3+sqrt2)/2

Answer 25

x^2 - 3x + (11/4 + 3/sqrt2)

Answer 26

where did you get the 11/4

Answer 27

?

Answer 28

(3-sqrt2)/2 x (3+sqrt2)/2
=1/4(9 - 3sqrt2 + 3sqrt2 -2)
= 1/4 x 7
=7/4 my first answer was incorrect

Answer 29

so we have x^2 - 3x +7/4

Answer 30

follow so far?

Answer 31

i see what your doing, where does the -2 come from though?

=1/4(9 - 3sqrt2 + 3sqrt2 -2)<<

Answer 32

oh wait nevermind got it

Answer 33

-sqrt2 x sqrt2 = -2

Answer 34

square root of 4 ))

Answer 35

should that be 2i?

Answer 36

no

Answer 37

ok

Answer 38

carry on

Answer 39

if we now multiply (x^2 - 3x +7/4) with (x -2i) we get a cubic, but not all the co-efficients are real. so there must be another root probably -2i

Answer 40

so still there is 3 roots?

Answer 41

can't be . if you have 2i as a root then there must be a conjugate root which would be -2i

Answer 42

alright i get a better idea of this now. but it's late, ill look at this AGAIN tomorrow. thanks

Answer 43

ok. just multiply (x+2i)(x-2i) and then multiply the result with our quadratic above

Answer 44

you'll get a polynomial with degree 4 and with real co-efficients