If we divide x^4+4x^3+2x^2+x+4 by x^2+3x, what will be the remainder?

If you're familiar with synthetic division, but not the extended form (which allows you easily compute the quotient/remainder when dividing a polynomial by another polynomial of degree greater than 1), then you can perform two steps of SD.

Instead of dividing by , first divide by , then by (since ). So we have

0   |   1    4    2    1    4
...  |         0    0    0    0
= = = = = = = = = = = =
...  |   1    4    2    1    4

which translates to



Ignoring the remainder term for now, the next round of SD yields

-3   |   1    4    2    1
...   |        -3   -3    3   
= = = = = = = = = =
...   |   1    1   -1    4

which translates to



Now, putting everything together, we have





which is to say the remainder upon dividing by is .