How do you evaluate #f(x)=-4x^3+3x-5# at x=2 using direct substitution and synthetic division?

1 Answer
Jul 21, 2018

Answer:

The remainder is #-31# and the quotient is #=-4x^2-8x-13#

Explanation:

Let's perform the synthetic division

#color(white)(aaaa)##2##|##color(white)(aaaa)##-4##color(white)(aaaaa)##0##color(white)(aaaaaa)##3##color(white)(aaaaaaa)##-5#

#color(white)(aaaaa)##|##color(white)(aaaa)##color(white)(aaaaaa)##-8##color(white)(aaaa)##-16##color(white)(aaaaa)##-26#

#color(white)(aaaaaaaaa)###_________

#color(white)(aaaaa)##|##color(white)(aaaa)##-4##color(white)(aaaa)##-8##color(white)(aaaa)##-13##color(white)(aaaaa)##color(red)(-31)#

The remainder is #-31# and the quotient is #=-4x^2-8x-13#

#(-4x^3+3x-5)/(x-2)=-4x^2-8x-13-31/(x-2)#

Apply the remainder theorem

When a polynomial #f(x)# is divided by #(x-c)#, we get

#f(x)=(x-c)q(x)+r#

Let #x=c#

Then,

#f(c)=0+r#

Here,

#f(x)=-4x^3+3x-5#

Therefore,

#f(2)=-4*2^3+3*2-5#

#=-32+6-5#

#=-31#