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

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Answer 1 · 2017-04-28T05:45:08

$f(-3) = 225$

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 $f(-3) = 225$

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