Question

llaa...

Answer 1

if 2 + sqrt 3

Answer 2

sorry

Answer 3

roots a + bsqrt(2) come in conjugate pairs

Answer 4

the reason why is because your polynomial has integer coefficients

Answer 5

( x - (a + bsqrt(2)) * ( x - ( a - bsqrt(2)) , expand this

Answer 6

so thats why it should be a root?

Answer 7

Where is it stated that the polynomial has integer coefficients?

Answer 8

im confused

Answer 9

Here is a perfectly good polynomial which has only the root \(2\sqrt{3} \).
\(x - 2\sqrt{3} = 0\)

Answer 10

so how is it a root?

Answer 11

@mathstudent55

Answer 12

Are there instructions to your question?

Answer 13

you can assume that the polynomial has rational coefficients , otherwise the question does not make sense

Answer 14

just what i put in the original question

Answer 15

are you sure @perl

Answer 16

im fairly certain

Answer 17

thank you :)

Answer 18

so what happens when you multiply
( x - ( 2 + sqrt(3))) * ( x - ( 2 - sqrt(3) ) ) ?

Answer 19

is that a graph?

Answer 20

@perl is correct.
Without that assumption, you may have only the root I showed above. The question is making the assumption that the coefficients of the polynomial are rational. Then as @perl stated above, the roots come in pairs of complex conjugate numbers.

Answer 21

so what does that mean?

Answer 22

that doesnt really answer my question..?

Answer 23

The question stated the root is \(2\sqrt 3\)
Since there must be two conjugate roots, the roots must be:
\(2 \sqrt 3\) and \(-2\sqrt 3\)

Answer 24

are 2sqrt3 and 2 + sqrt3 different?

Answer 25

A polynomial with roots a and b has the form:
(x - a)(x - b)
Replace a and b with the two roots, and multiply it out with FOIL.
That will give you your polynomial.

Answer 26

Yes.
\(2 + \sqrt 3\) means add 2 and \(\sqrt 3\) which means approx 3.73
\(2 \sqrt 3\) means multiply 2 by \(\sqrt 3\) which means approx. 3.46

Answer 27

mine is 2 + sqrt 3

Answer 28

i just dont uinderstand what the answer would be.. what i need to answer..

Answer 29

Above, in the post, you have 2 sqrt 3 with no addition sign.

Answer 30

i know and then i changed it right below my original question

Answer 31

Is there also an i, for complex number missing?

Answer 32

no there is no i

Answer 33

Oh, I see. I missed that, sorry.

Answer 34

its okay :)

Answer 35

Wait I just calculated it again.
perl was correct above.
Just do:
( x - ( 2 + sqrt(3))) * ( x - ( 2 - sqrt(3) ) )
Multiply it out using FOIL

Answer 36

then that will give me what?

Answer 37

\([ x - ( 2 + \sqrt 3)] \times [ x - ( 2 - \sqrt 3 ) ]\)

\(=x^2 - x(2 - \sqrt3) - x(2 + \sqrt3) + (2 + \sqrt 3)(2 - \sqrt 3)\)

\(=x^2 - 2x + x\sqrt3 - 2x -x \sqrt3 + 4 - 3 \)

\(=x^2 - 4x + 1\)