Finding Remaining Roots

[thunc14]

So here's the problem: If you have the polynomial x^5-3x^4+6x^3+122x^2-619x+693 and are given that one of the roots is 2-5i, which technique do you NOT use when finding the remaining roots?

A) Conjugates
B) Factoring
C) Polynomial Long Division
D) Synthetic Division
E) All of the above are used

So I graphed this in desmos and plugged it into wolfram alpha to see what the remaining roots were before I got started. I do know that conjugates are used due to the complex root theorem. Factoring by grouping doesn't work (I think), so the rational root theorem seemed like the next best place to go. From what I have seen, synthetic division is done with the rational root theorem, but I dont think that that makes polynomial long division incorrect to do? Just two different methods to get to the same answer is it not? When you do the division, you get (x-1)(x^4-2x^3+4x^2+126x-463). So the division has essentially factored out (x-1), so then you have technically applied factoring as a technique?

This question is making my head spin a bit, since the easiest thing to do would be just to use the complex root theorem and plug the equation into a graphing calculator and get your zeros.

Any advice for this one? I went with E

 

[Deleted member 4993]

So here's the problem: If you have the polynomial x^5-3x^4+6x^3+122x^2-619x+693 and are given that one of the roots is 2-5i, which technique do you NOT use when finding the remaining roots?

A) Conjugates
B) Factoring
C) Polynomial Long Division
D) Synthetic Division
E) All of the above are used

So I graphed this in desmos and plugged it into wolfram alpha to see what the remaining roots were before I got started. I do know that conjugates are used due to the complex root theorem. Factoring by grouping doesn't work (I think), so the rational root theorem seemed like the next best place to go. From what I have seen, synthetic division is done with the rational root theorem, but I dont think that that makes polynomial long division incorrect to do? Just two different methods to get to the same answer is it not? When you do the division, you get (x-1)(x^4-2x^3+4x^2+126x-463). So the division has essentially factored out (x-1), so then you have technically applied factoring as a technique?

This question is making my head spin a bit, since the easiest thing to do would be just to use the complex root theorem and plug the equation into a graphing calculator and get your zeros.

Any advice for this one? I went with E

I would have gone with E too.

However, preferred wording of E would be to replace "are" with "can".

I do not quite know - how is the term "factoring" is used in this context!

 

[thunc14]

I would have gone with E too.

However, preferred wording of E would be to replace "are" with "can".

I do not quite know - how is the term "factoring" is used in this context!

That's exactly my point as well! This has to be the 5th problem or so that I've posted where the reaction is the same, the context, question wording, and answer choices are extremely unclear. Incredibly frustrating

 

[Cubist]

This post doesn't help in finding an answer (I too think that E might be correct) but...

OP seems to have made a couple of typing mistakes (there were a lot of numbers in there!)...

So here's the problem: If you have the polynomial x^5-3x^4+6x^3+122x^2-619x+693

Probably should be x^5 - 3x^4 + 6x^3 + 122x^2 - 619x + 493 ?

When you do the division, you get (x-1)(x^4-2x^3+4x^2+126x-463)

and this should be (x-1)(x^4 - 2x^3 + 4x^2 + 126x - 493) ?


NOTE when you're given a root I'd recommend that you use that information ASAP. The given 2-5i AND its conjugate 2+5i are both roots since the polynomial has real coefficients. So it should divide by (x - {2-5i}) (x - {2+5i}) or multiplied out x^2 -4*x + 29. Then you'd be left with a cubic.

(This is how I spotted the typos, because your given poly would not divide by the above!)