modular reduction of polynomials

[caledonia]

I have seen asserted that for polynomials f and h, f divides h implies that fbar divides hbar, where "bar" indicates reduction modulo p, a prime number say.

 

But when I test this with sample polynomials, it does not seem to be true. For example modulo 3 and f = 2x + 1, h = 6x² + 7x + 2, then fbar = 2x + 1, hbar = x + 2.

What is wrong here please ?

 

[mathematic]

The quotient of h over f is 3x+2 = 2(modulo 3). 2fbar=4x+2=x+2(modulo 3), which is hbar.

[caledonia]

Thank you. But how about this case (mod 3) :

f = 11x + 2, g = 5x + 7, then fg = 55x² + 87x + 14, fbar = 2x + 2, gbar = 2x + 1, (fg)bar = x² + 2.

How can fbar (or gbar) divide (fg)bar ?

[mathematic]

fbar x gbar = 4x²+6x+2=x²+2 (mod 3)

 

I don't know why you had a problem with this.