How do you use synthetic substitution to evaluate f(-3) for #f(x)=x^4-4x^3+2x^2-4x+6#?

1 Answer
Apr 28, 2017

Answer:

#f(-3) = 225#

Explanation:

Given: #f(x) = x^4 - 4x^3 +2x^2 - 4x + 6#

#f(-3) = (-3)^4 - 4(-3)^3 + 2(-3)^2 - 4(-3) +6 = 225#

According to the Remainder Theorem: #(f(x))/(x- -3) = (f(x))/(x+3) = 225#

Synthetic Division , where #x+3 = 0" or " x = -3:#

terms:#" "x^4" "x^3" "x^2" "x" constant"#

#ul(-3)| " "1" "-4" "2" "-4" "6#
#" "ul(+" "-3" "21" "-69" "219)#
#" "1" "-7" "23" "-73" "225#

terms:#" "x^3" "x^2" "x" constant, remainder"#

This means

#(x^4-4x^3+2x^2-4x+6)/(x+3) = x^3 - 7x^2 +23x -73 +225/(x+3)#

The remainder of synthetic division (the last value) is #f(-3) = 225#