How do you divide #x^4-2x^2+17x# by #x-2#?

1 Answer
Nov 22, 2015

Use either synthetic division or polynomial long division to get
#x^3+2x^2+2x+21# with remainder #84#

Explanation:

Remember to include the #0# value coefficients for #x^3# and the constant.

synthetic division

#{: (,,color(brown)(x^4),color(brown)(x^3),color(brown)(x^2),color(brown)(x^1),color(brown)(x^0)), (,"|",1,0,-2,+17,0), (,"|",,2,4,4,84), ("----",,"----","----","----","----","----"), (xx(color(cyan)(+2)),"|",color(blue)(1),color(blue)(2),color(blue)(2),color(blue)(21),color(red)(84)), (,,color(brown)(x^3),color(brown)(x^2),color(brown)(x^1),color(brown)(x^0),color(brown)("Rem")) :}#
...also remember that for synthetic division (by a monic binomial) the #color(cyan)("multiplier")# is the negative of the constant from the divisor.

polynomial long division

#{: (,,x^3,+2x^2,+2x,21,), (,,"----","----","----","----","----"), (x-2,")",x^4,,-2x^2,+17x,), (,,x^4,-2x^3,,,), (,,"----","----",,,), (,,,2x^3,-2x^2,,), (,,,2x^3,-4x^2,,), (,,,"----","------",,), (,,,,2x^2,+17x,), (,,,,2x^2,-4x,), (,,,,"----","------",), (,,,,,21x,), (,,,,,21x,-84), (,,,,,"----","-----"), (,,,,,,84) :}#