Question

P(x) with rational coefficients has the given roots (i and 7 + 8i) Find two additional roots?

Answer 1

Hint: the complex roots of a polynomial \(P(x)\) with rational coefficients come in pair as \(a+bi\) and \(a-bi\)

Answer 2

I don't really understand that

Answer 3

I know how to find roots but how do i go backwards to finding the equation given the roots?

Answer 4

Oh do you mean the two roots would be put into form as:

(x + i)(x - i) and (x + (7 +8i)) (x - (7 + 8i))?

Answer 5

close.
one of the roots is \(i\) therefore its conjugate pair is \(-i\)
the other root is \(7+8i\) so its conjugate pair is \(7-8i\)

Answer 6

So (x + i)(x - i) isn't correct?

I see how (x + (7 + 8i)) (x - (7 - 8i)) is correct but what do i do with the simplified root that get when i multiply?

Answer 7

the roots are \(i,\; -i,\; 7+8i,\;7-8i\).
If you are reconstructing \(P(x)\), then \[P(x)=(x-i)(x-(-i))(x-(7+8i))(x-(7-8i))\]

Answer 8

Ah ok. And once P(x) is known to find two additional roots use rational root theorem and synthetic division?

Answer 9

there is no need to use rational root and/or sythetic division for this problem. \(P(x)\) was reconstructed from the roots, so you knew the roots beforehand.

Answer 10

The question i needed to answer originally though asked me for 2 additional roots given the roots i and 7 + 8i.

Answer 11

The exact question is "A polynomial function P(x) with rational coefficients has the given roots. Find two additional roots of P(x)=0."

Answer 12

exactly. There is no need to write \(P(x)\). You only need to give the conjugate pair of each complex root.
The conjugate pair of \(i\) is \(-i\), so you have \(-i\) as one of the two roots asked.

Answer 13

Oh i see now... One last though, in the P(x) = .... you typed above in each parentheses it was always (x - the root or the roots opposite) is it always x MINUS? Is that just how its done?

Answer 14

Oh and I'm sorry to keep asking questions separate from my original; but lets say one of the given roots is the square root of 10, would the inverse be i square root 10?

Answer 15

that's how reconstruction of \(P(x)\) is done.
if one of the roots is \(\sqrt{10}\) which you can write as \(\sqrt{10}+0i\), its conjugate is \(\sqrt{10}-0i\)

Answer 16

So after you get the reconstruction of P(x) and simplify it that's the final answer?

Answer 17

yes. but you only do that when \(P(x\) is asked.

Answer 18

Well thank you very much for your help, i think i can figure these out now.

Answer 19

if you're up to it, you can write the polynomial \(P(x)\) with rational coefficients whose two roots are \(i\) and \(7+8i\).