Use synthetic division to solve: #(x^2+7x-1)# divided by #(x+1)# ?

1 Answer
George C.
Feb 18, 2018

#(x^2+7x-1)/(x+1) = x+6-7/(x+1)#

Explanation:

We start by writing the coefficients of the dividend inside an L shape and the zero associated with the divisor just outside:

#-1color(white)(" ")"|"color(white)(" ")1color(white)(" ")7color(white)(" ")color(black)(-1)#
#color(white)(-1" ")"|"underline(color(white)(" "1" "7" "-1)#

Carry the first coefficient from the dividend down to below the line:

#-1color(white)(" ")"|"color(white)(" ")1color(white)(" ")7color(white)(" ")color(black)(-1)#
#color(white)(-1" ")"|"underline(color(white)(" "1" "7" "-1))#
#color(white)(-1" ")color(white)("|")color(white)(" ")1#

Multiply this first coefficient of the quotient by the test zero and write it in the second column:

#-1color(white)(" ")"|"color(white)(" ")1color(white)(" "-)7color(white)(" ")color(black)(-1)#
#color(white)(-1" ")"|"underline(color(white)(" "1" ")-1color(white)(" "-1))#
#color(white)(-1" ")color(white)("|")color(white)(" ")1#

Add up the second column and write the sum as the next term of the quotient:

#-1color(white)(" ")"|"color(white)(" ")1color(white)(" "-)7color(white)(" ")color(black)(-1)#
#color(white)(-1" ")"|"underline(color(white)(" "1" ")-1color(white)(" "-1))#
#color(white)(-1" ")color(white)("|")color(white)(" ")1color(white)(" "-)6#

Multiply this second coefficient of the quotient by the test zero and write it in the third column:

#-1color(white)(" ")"|"color(white)(" ")1color(white)(" "-)7color(white)(" ")color(black)(-1)#
#color(white)(-1" ")"|"underline(color(white)(" "1" ")-1color(white)(" ")color(black)(-6)#
#color(white)(-1" ")color(white)("|")color(white)(" ")1color(white)(" "-)6#

Add up the third column to give the remainder:

#-1color(white)(" ")"|"color(white)(" ")1color(white)(" "-)7color(white)(" ")color(black)(-1)#
#color(white)(-1" ")"|"underline(color(white)(" "1" ")-1color(white)(" ")color(black)(-6)#
#color(white)(-1" ")color(white)("|")color(white)(" ")1color(white)(" "-)6color(white)(" ")color(red)(-7)#

Reading off the coefficients, we have found:

#(x^2+7x-1)/(x+1) = x+6-7/(x+1)#

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