How do you use synthetic division to divide #( 6x^3 - x^2 - 7x + 5 ) / ( x - 3 )#?

1 Answer
Sep 20, 2015

Answer:

#(6x^3-x^2-7x+5)/(x-3) = color(blue)(6x^2+17x+44)+color(red)(137/(x-3))#

Explanation:

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

#color(white)(1)|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#

Step 2. Put the divisor at the left.

#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#

Step 3. Drop the first coefficient of the dividend below the division symbol.

#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X)#
#" "stackrel("—————————————)#
#color(white)(1)|color(red)6#

Step 4. Multiply the result by the constant, and put the product in the next column.

#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X1)18#
#" "stackrel("—————————————)#
#color(white)(1)|color(blue)6#

Step 5. Add down the column.

#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X1)18" "color(white)(1)51color(white)(X1)132#
#" "stackrel("—————————————)#
#color(white)(1)|color(blue)6" "color(red)(color(white)(X)17#

Step 6. Repeat Steps 4 and 5 until you can go no farther.

#3|6" "-1color(white)(X)-7" "" "5#
#color(white)(1)|" "color(white)(X1)18" "color(white)(1)51color(white)(X1)132#
#" "stackrel("—————————————)#
#color(white)(1)|color(blue)(6)" "color(white)(X)17" "color(white)(1)44" "color(white)(1)color(red)(137)#

Answer:

#(6x^3-x^2-7x+5)/(x-3) = 6x^2+17x+44+ (137)/(x-3)#

Check:

#(x-3)(6x^2+17x+44+137/(x-3)) = (x-3)(6x^2+17x+44)+137 = 6x^3+17x^2+44x-18x^2-51x-132+137= 6x^3-x^2-7x+5#

It works!