The given: Dividend #(x^4-11x^3+15x^2+40x-5)#

and Divisor #(x-5)#

Synthetic Division:

Start with the dividend,

Arrange the terms from highest to lowest degree:

#x^4#---- #x^3#----#x^2#----#x^1#----#x^0#

Use the numerical coefficients only and the divisor #x-5# be equated to zero.

that is #x-5=0# solving for x: results to #x=+5# the trial divisor.

1 s t Line:#1 \\ -11 \\ +15 \\+40 \ \ -5# use trial divisor=#+5#

2nd Line:#0 \ \ +5 \ \-30\ \-75\\-175#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

3rd Line:#1 \ \ -6 \ \ -15 \ \-35\\-180#

Bring down the first numerical coefficient 1. This number will be multiplied by the trial divisor 5 result is 5 which will be written at the 2nd column under #-11#. Perform Algebraic addition using #-11# and #5# and result is #-6# located at the second column of 3rd line.

Next #-6# be multiplied by the trial divisor #+5#, result is #-30# to be written at the 3rd column under the #+15#. Perform Algebraic addition using #+15# and #-30# and result is #-15# located at the 3rd column of 3rd line.

Next #-15# be multiplied by the trial divisor #+5#, result is #-75# to be written at the 4th column under the #+40#. Perform Algebraic addition using #+40# and #-75# and result is #-35# located at the 4th column of 3rd line.

Next #-35# be multiplied by the trial divisor #+5#, result is #-175# to be written at the 5th column under the #-5#. Perform Algebraic addition using #-5# and #-175# and result is #-180# located at the 5th column of 3rd line. And this is the REMAINDER.

Write the answer:

#(DIVIDEND)/(DIVISOR)=QUOTIENT+(REMAINDER)/(DIVISOR)#

#(x^4-11x^3+15x^2+40x-5)/(x-5)=x^3-6x^2-15x-35+(-180)/(x-5)#

Have a nice day!!! from the Philippines..