How do you use synthetic substitution to find x=-3 for # f(x)= -x^4+x^3-x+1#?

1 Answer
sankarankalyanam
Jan 30, 2018

#f(x) =color(blue)( f(-3) = -104)#

Quotient #color(green)(Q = -x^3 + 4x^2 - 12x + 35)#, Remainder #color(red)(R = (-104)/(x+3))#

Explanation:

Direct Substitution : #f(x) = f(-3) = -(-3^4) + (-3)^3 -(-3) + 1#

#f(-3) = = -81 - 27 +3 +1 = -104#

Synthetic Division :

#color(white)(aa)-3color(white)(aa)|color(white)(aa)-1color(white)(aaa)1color(white)(aaaaa)0color(white)(aa)-1color(white)(aaaaa)1#
#color(white)(aaaaaaaa)|color(white)(aaa)darrcolor(white)(aaa)3color(white)(a)-12color(white)(aaaa)36color(white)(a)-105#
#color(white)(aaaaaaa)------------#
#color(white)(aaaaaaaaaaa)-1color(white)(aaaa)4color(white)(a)-12color(white)(aaaa)35color(white)(a)-104#

Quotient #color(green)(Q = -x^3 + 4x^2 - 12x + 35)#, Remainder #color(red)(R = (-104)/(x+3))#

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