Question

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

Answer 1

`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))`