How can the Rational Roots Theorem help with factoring?

[Miike012]

Homework Statement

I added attachment...

 

[symbolipoint]

The document shows too small and too blurry, even after I change the screen magnification;actually, the magnification seems to have no effect, even no change at 200%. It appears to be something about polynomial or synthetic division.

 

[Miike012]

I added another attachment... I made it bigger...
if you can't read it... the problem is...

What is the HCF of
x^4 + 3x^3 +12x -16 and x^3 -13x+12

 

[Miike012]

I have another quick question if you don't mind answering?

The question is what is the HCF of 2x^3 + 4x^2 - 7x -14 and 6x^3 - 10x^2 -21x +35

If I multiply 3 by the first equation then subtract from the second equation I get...

6x^3 - 10x^2 -21x +35 - 6x^3 - 12x^2 + 21x +42 = -22x^2 + 77 = -11(2^2-7)

HCF = (2^2-7)
How do I know to stop at -22x^2 + 77? And why isn't -11 included into the HCF? Is it because there isn't a simple factor in the two originals therefore there will not be a simple factor in the HCF?

 

[symbolipoint]

Mike012,
This may be too simple compared to how you want to factor, but I would try some sense from the Rational Roots Theorem. Initially this would check for linear factors, but it would still give enough results for degree 2 or degree 3; there may be either a degree 2 which itself might not be factorable into linears, or there may be a degree 2 which is composed of two linears.