Cubic polynomial unable to be factored nicely

In summary, the given expression 6x3 - 3x2 + 12 cannot be factored using the rational root theorem and is likely a typo in a past exam.
  • #1
anniecvc
28
0
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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  • #2
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.
 

What is a cubic polynomial?

A cubic polynomial is a mathematical expression that contains a variable raised to the power of 3, such as x^3, and may also contain other terms with lower powers of the variable.

What does it mean for a cubic polynomial to be unable to be factored nicely?

When a cubic polynomial cannot be factored nicely, it means that it cannot be expressed as the product of simpler polynomials with integer coefficients. This may be due to the polynomial having irrational or complex roots.

How do you know if a cubic polynomial cannot be factored nicely?

A cubic polynomial cannot be factored nicely if it cannot be factored using basic algebraic techniques, such as grouping or difference of squares. In other words, there is no simple way to write the polynomial as a product of simpler polynomials.

Why is it important to be able to factor cubic polynomials?

Factoring cubic polynomials is important because it allows us to solve equations involving these polynomials and find their roots. It also helps us understand the behavior and properties of cubic functions, which are commonly used in modeling real-world phenomena.

Is there a general formula for factoring cubic polynomials?

No, there is no general formula for factoring cubic polynomials. However, there are some techniques that can be used to factor certain types of cubic polynomials, such as those with rational roots or those that can be written as a difference of cubes.

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