How do you use synthetic substitution to find x=-3 for f(x)= -x^4+x^3-x+1f(x)=x4+x3x+1?

1 Answer
Jan 30, 2018

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

Quotient color(green)(Q = -x^3 + 4x^2 - 12x + 35)Q=x3+4x212x+35, Remainder color(red)(R = (-104)/(x+3))R=104x+3

Explanation:

Direct Substitution : f(x) = f(-3) = -(-3^4) + (-3)^3 -(-3) + 1f(x)=f(3)=(34)+(3)3(3)+1

f(-3) = = -81 - 27 +3 +1 = -104f(3)==8127+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)1aa3aaaa1aaa1aaaaa0aa1aaaaa1
color(white)(aaaaaaaa)|color(white)(aaa)darrcolor(white)(aaa)3color(white)(a)-12color(white)(aaaa)36color(white)(a)-105aaaaaaaaaaaaaa3a12aaaa36a105
color(white)(aaaaaaa)------------aaaaaaa
color(white)(aaaaaaaaaaa)-1color(white)(aaaa)4color(white)(a)-12color(white)(aaaa)35color(white)(a)-104aaaaaaaaaaa1aaaa4a12aaaa35a104

Quotient color(green)(Q = -x^3 + 4x^2 - 12x + 35)Q=x3+4x212x+35, Remainder color(red)(R = (-104)/(x+3))R=104x+3