How do you evaluate #f(x)=-3x^3+7x^2-4x+8# at x=3 using direct substitution and synthetic division?

1 Answer
Aug 10, 2017

Answer:

The remainder is #color(red)(-22)# and the quotient is #=-3x^2-2x-10#

Explanation:

Let's perform the synthetic division

#color(white)(aaaa)##3##color(white)(aaaaaa)##|##color(white)(aa)##-3##color(white)(aaaaaaa)##7##color(white)(aaaaaa)##-4##color(white)(aaaaaaa)##8#
#color(white)(aaaaaaaaaaaa)##------------#

#color(white)(aaaa)##color(white)(aaaaaaa)##|##color(white)(aaaa)##color(white)(aaaaaa)##-9##color(white)(aaaaaa)##-6##color(white)(aaaaa)##-30#
#color(white)(aaaaaaaaaaaa)##------------#

#color(white)(aaaa)##color(white)(aaaaaaa)##|##color(white)(aaa)##-3##color(white)(aaaaa)##-2##color(white)(aaaaa)##-10##color(white)(aaaaa)##color(red)(-22)#

The remainder is #color(red)(-22)# and the quotient is #=-3x^2-2x-10#

Therefore,

#(-3x^3+7x^2-4x+8)=(-3x^2-2x-10)-22/(x-3)#

Also,

#f(3)=-3*(3)^3+7*(3)^2-4*(3)*8=-81+63-12+8=-93+71=-22#