How do you use the remainder theorem and Synthetic Division to find the remainders in the following division problems #x^3 - 2x^2 + 5x - 6# divided by x - 3?

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Answer 1 · 2016-12-15T14:30:09

The quotient is #=(x^2+x+8)# and the remainder is #=18#

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

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

Let #x=c#, then #f(c)=r#

Here, #f(x)=x^3-2x^2+5x-6# and #c=3#

Therefore,

#f(3)=3^3-2*3^2+5*3-6=27-18+15-6=18#

The remainder is #=18#

Let's do the long division

#color(white)(aaaa)##x^3-2x^2+5x-6##color(white)(aaaa)##∣##x-3#

#color(white)(aaaa)##x^3-3x^2##color(white)(aaaaaaaaaaaa)##∣##x^2+x+8#

#color(white)(aaaaa)##0+x^2+5x#

#color(white)(aaaaaaaa)##x^2-3x#

#color(white)(aaaaaaaaa)##0+8x-6#

#color(white)(aaaaaaaaaaa)##+8x-24#

#color(white)(aaaaaaaaaaaaaa)##0+18#

The quotient is #=(x^2+x+8)# and the remainder is #=18#