I like to just long divide the coefficients, not forgetting to include 0's for missing powers of x. In addition, I'm not keen on fractions, so I will choose to multiply the numerator by 2 first, then divide by 2 at the end.
2×(4x4+4x3−8x+2)=8x4+8x3+0x2−16x+4
![enter image source here]()
This process is similar to long division of decimal numbers.
Write the dividend (8,8,0,−16,4) under the bar and the divisor (2,−3,1) to the left of the bar.
Choose the first term 4 of the quotient so that when multiplied by the divisor it matches the leading term of the dividend.
Write the product (8,−12,4) under the dividend and subtract it to get a remainder. Bring down the next term −16 of the dividend alongside it to give the running remainder.
Choose the next term 10 of the quotient so that when multiplied by the divisor it matches the first term of the running remainder.
Write the product (20,−30,10) under the running remainder and subtract it to get the new running remainder. Bring down the next term 4 of the dividend alongside it.
Choose the final term 13 of the quotient as before, write down the product (26,−39,13) and subtract it to give the final remainder (13,−9).
So:
8x4+8x3−16x+4=(2x2−3x+1)(4x2+10x+13)+(13x−9)
Dividing by 2:
4x4+4x3−8x+2
=(2x2−3x+1)(2x2+5x+132)+(132x−92)
In other words:
4x4+4x3−8x+22x2−3x+1
=(2x2+5x+132)+132x−922x2−3x+1
