How do you use synthetic division to show that x=2 is a zero of #x^3-7x+6=0#?

1 Answer
Jan 25, 2017

Answer:

Please see below.

Explanation:

To divide #x^3-7x+6=x^3+0x^2-7x+6# by #x-2# (as we have to show #x=2#, a zero).

One Write the coefficients of #x# in the dividend inside an upside-down division symbol.

#color(white)(1)|color(white)(X)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#

Two As #x=2# we put #2# at the left.

#2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(XX)#
#" "stackrel("—————————————)#

Three Drop the first coefficient of the dividend below the division symbol.

#2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(X)#
#" "stackrel("—————————————)#
#color(white)(x)|color(white)(x)color(red)1#

Four Multiply the result by the constant, and put the product in the next column.

#2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(XX)2#
#" "stackrel("—————————————)#
#color(white)(x)|color(white)(x)color(blue)1#

Five Add down the column.

#2|color(white)(x)1" "color(white)(x)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(XX)2#
#" "stackrel("—————————————)#
#color(white)(x)|color(white)(x)color(blue)1color(white)(X11)color(red)2#

Six Repeat Steps Four and Five until you can go no farther.

#2|color(white)(x)1" "color(white)(X)0color(white)(XX)-7" "" "6#
#color(white)(x)|" "color(white)(XX)2color(white)(xxxxx)4color(white)(X)-6#
#" "stackrel("—————————————)#
#color(white)(x)|color(white)(x)color(blue)1color(white)(X11)color(red)2color(white)(XX)color(red)-3color(white)(XXX)color(red)0#

Hence, Quotient is #x^2+2x-1# and remainder is #0#.

As remainder is #0#, #x=2# is a zero of #x^3-7x+6#.