Synethetic substitution is the process of using synthetic division to find #g(x)#. To find #g(-3)#, use synthetic division with a divisor of #-3#. The remainder equals #g(-3)#
Use the coefficients of the polynomial as the dividend. Note that the #x^2# term has a coefficient of zero.
#g(x)=color(red)1x^3+color(red)0x^2color(red)(-8)x+color(red)6#
#-3| color(red)1color(white)(aaaa)color(red)0color(white)(aa)color(red)(-8)color(white)(aaa)color(red)6#
#color(white)(aaa^2)darrcolor(white)(aaaaaaaaaaaaaaaa)#Pull down the #color(red)1#
#color(white)(aaa^2a)color(magenta)1#
#-3| color(red)1color(white)(aaaa)color(red)0color(white)(aa)color(red)(-8)color(white)(aaa)color(red)6#
#color(white)(aaa^2)darrcolor(white)(a^2)color(blue)(-3)color(white)(aaaaaaaaaa)# Multiply #-3 * color(magenta)1=color(blue)(-3)#
#color(white)(aaaa^2color(magenta)1color(white)(aa)color(magenta)(-3)color(white)(aaaaaaaaaaa)#Add #color(red)0 +color(blue)(-3) =color(magenta)(-3)#
Repeat the process of multiplying by #-3# and adding to the coefficient.
#-3| color(red)1color(white)(aaaa)color(red)0color(white)(aa)color(red)(-8)color(white)(aaa)color(red)6#
#color(white)(aaa^2)darrcolor(white)(a^2)color(blue)(-3)color(white)(aaa)color(blue)9color(white)(aa)color(blue)(-3)#
#color(white)(aaaa^2color(magenta)1color(white)(aa)color(magenta)(-3)color(white)(aaa)color(magenta)1color(white)(aa)aacolor(orange)3#
The remainder #color(orange)3= g(-3)#
