How do you use synthetic division to divide #3x^3+4x^2-7x+1# by #3x-2#?

2 Answers
Jul 30, 2015

Answer:

#color(red)((3x^3 +4x^2-7x+6)/(3x-2) = x^2+2x-1-1/(3x-2))#

Explanation:

We use a slightly modified table when the coefficient of #x# does not equal #1#. Note the extra lines.

Step 1. Write only the coefficients of #x# in the dividend inside an upside-down division symbol.

#|3" "4" "-7" " " " "1#
#|color(white)(1)#
#stackrel("——————————————)#

Step 2. Put the divisor at the left.

#" "" "|3" "4" "-7" " " " "1#
#" "color(red)(2)color(white)(1)|#
#" "stackrel("——————————————)#

Step 3. Write the coefficient of #x# below the division line

#" "" "|3" "4" "-7" " " " "1#
#" "2|" "color(white)(1)2 " "" "4" "-2#
#" "stackrel("——————————————)#
#" "color(white)(1)|#
#color(red)(/3)color(white)(1)|#

Step 4. Drop the first coefficient of the dividend below the division symbol.

#" "" "|3" "4" "-7" " " " "1#
#" "2|" "color(white)(1)2 " "" "4" "-2#
#" "stackrel("——————————————)#
#" "color(white)(1)|color(red)(3)#
#/3color(white)(1)|#

Step 5. Divide the dropped value by the coefficient of #x# and place the result in the row below.

#" "" "|3" "4" "-7" " " " "1#
#" "2|" "color(white)(1)2 " "" "4" "-2#
#" "stackrel("——————————————)#
#" "" "|3#
#/3color(white)(1)|color(red)(1)#

Step 6. Multiply the result by the constant, and put the product in the next column.

#" "" "|3" "4" "-7" " " " "1#
#" "2|" "color(white)(1)color(red)(2)#
#" "stackrel("——————————————)#
#" "" "|3#
#/3color(white)(1)|1#

Step 7. Add down the column.

#" "" "|3" "4" "-7" " " " "1#
#" "2|" "color(white)(1)2#
#" "stackrel("——————————————)#
#" "" "|3" "color(red)(6)#
#/3color(white)(1)|1#

Step 8. Repeat Steps 5, 6, and 7 until you can go no farther.

#" "" "|3" "4" "-7" " " " "1#
#" "2|" "color(white)(1)2 " "" "4" "-2#
#" "stackrel("——————————————)#
#" "" "|3" "6" "-3" "color(red)(-1)#
#/3color(white)(1)|1" "2" "-1#

#(3x^3 +4x^2-7x+6)/(3x-2) = x^2+2x-1-1/(3x-2)#

Check:

#(3x-2)( x^2+2x-1-1/(3x-2)) = (3x-2)(x^2+2x-1)-1#

#= 3x^3+6x^2-3x-2x^2-4x+2-1 = 3x^3+4x^2-7x +1#

Jul 30, 2015

Answer:

#(3x^3+4x^2-7x+1) div (3x-2)#
#color(white)("XXXX")##= x^2+2x-1##color(white)("XXXX")#Remainder: -1

Explanation:

Note that this is simply an alternate approach to Ernest Z's answer. Some people may find one approach easier to understand than the other.

Set up as standard long division:
enter image source here

#3x# " goes into " #3x^3##color(white)("XXXX")##rarr x^2# times:
enter image source here

Multiply #3x-2# by #x^2# and write the product below the line:
enter image source here

Subtract:
enter image source here

"Bring down" the #-7x#
enter image source here

#3x# " goes into " #6x^2##color(white)("XXXX")##rarr 2x# times
...and so on...
enter image source here

The Remainder of #(-1)# may be simply noted as a remainder or written as a fraction #(-1/(3x-2))_#