Cubic polynomial unable to be factored nicely

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SUMMARY

The cubic polynomial 6x³ - 3x² + 12 cannot be factored nicely using the rational roots theorem, as confirmed by synthetic division which yields no rational roots. The polynomial simplifies to 3[ x²(2x-1) + 4], but further attempts to factor reveal one irrational and two complex roots when analyzed with Wolfram Alpha. This indicates that the polynomial is not factorable in a straightforward manner, likely due to a typographical error in the original exam question.

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anniecvc
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Found this on a test for an integrated algebra 2 high school math class!

Factor completely.

6x3 - 3x2 + 12

The Attempt at a Solution



3( 2x3 - x2 + 4) eq.1

At this point I checked for rational roots using the rational roots theorem and synthetically dividing. I got nothing.

Although this doesn't get us anywhere x2 can be factored out of the first two terms:

3[ x2(2x-1) +4)] eq. 2

I then plugged this guy into Wolfram and it is giving me 1 nasty irrational and 2 nasty complex roots. I'm convinced this is NOT "nicely" factorable. Does anyone have any insights?
 
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You're correct that it can't be factorized using the rational root theorem, so as such, I wouldn't go any further with it. Since it was in a past exam, I'm sure it was a typo that they ended up correcting for the students on the day.
 

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