How do i factor 2x^3-4x^2+7x-27

Question

I have gotten the function down to 2x^2(x-2)+(7x-27)

Answer 1

Riley,

First, I would like to ask if you copied the polynomial correctly. 

If you did, then I will make a few comments:

• There are always several correct ways to solve these: plug-in guess and check, long division, synthetic division, tricks, graphically, etc.
• You can't quite use factoring by grouping here (because you can see that the second group will not have an x-2 in common), so don't factor out the 2x². But, this shows that you're thinking! That is something that you want to always check first with polynomials of higher degrees. I check for that automatically, and whether or not anything can be factored of the whole expression. Nothing can here, so we move on.
• With polynomials, finding factors involves finding zeros (the x-intercepts) of the function set equal to y, because all factors will be x-intercepts. Have you learned the p/q rule (the Rational Zeros Theorem)? For the given example, if any rational  factors exist, it would be one of the possible p/q. For that, you have to make a list: 
1. What are the factors of the last number, 27? There are 1, 3, 9, 27
2. What are the factors of the coefficient in the first term?  1, 2
3. Now divide all the top numbers with all the bottom ones to achieve possible factors, the p/q:

                     +/-1, +/-3, +/-9, +/-27, +/-(1/2), +/-(3/2), +/-(9/2), +/-(27/2)

• You could plug each into the expression first, to see if any will make the whole thing 0. Since every 1st and 3rd term changes sign when you plug a negative, you can get rid of all the negative values from the above list because they will never make that equal to zero.

 

Sadly, none of those work, so that indicates that there are no rational x-intercepts. In fact, it looks like there is only a irrational solution.

• If you are allowed to use a graphing calculator on tests and such, I would start there to look where the graph crosses.
• I went ahead and graphed it. With an appropriate viewing window, it shows that the graph crosses the x-axis once and zooming in further I got an estimate to three decimal places of 2.625

Therefore, with all this information, I'd have to (relatively confidently) conclude that this expression can't be factored. I'm not really sure why they gave you something that can't be factored to factor. Anyway, good luck. I hope this answers some doubts :-)

Answer 2

Suppose that           f(x) = 2x³ - 4x² + 7x - 27

 

 

To factor f(x), we need to examine the first and last terms of f(x).   What are its factors?  The factors of 2 are 1 and 2.  The factors of 27 are 1, 3, 9, and 27. 

 

According to the rational root theorem, the possible roots are the ratio of the factors of the last term to the first term.  This means the possible roots are

 

c = ±1, ±3 , ±9, ±27 , ±1/2 , ±3/2 , ±9/2, and ±27/2

 

where a possible factor is (x - c). If we divide f(x) by (x - c), the remainder must be zero.

 

Another way is to evaluate f(x) when x=c to see if f(x) becomes zero, and get your factor that way.

 

 

This process takes a lot of trial and error.  I would start with picking roots that have a high chance of working in your favor.

 

 

One last method to try is to use the graphing calculator to find all the zeros of f(x).  If x is an irrational number when     f(x) is zero, then f(x) cannot be factored.