How do you long divide #( (2x^3) - (5x^2) + (4x) - (4) ) div(x - 2) #?

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Answer 1 · 2016-12-04T15:04:54

The remainder is #=0# and the quotient is #=2x^2-x+2#

Let's do the long division

#color(white)(aaaa)##2x^3-5x^2+4x-4##∣##x-2#

#color(white)(aaaa)##2x^3-4x^2##color(white)(aaaaaaaa)##∣##2x^2-x+2#

#color(white)(aaaaaa)##0-x^2+4x#

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

#color(white)(aaaaaaaaaaa)##0+2x-4#

#color(white)(aaaaaaaaaaaaa)##+2x-4#

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

so, the remainder is #0# and the quotient is #2x^2-x+2#

You can also use the remainder theorem to see that the remainder is #=0#

If #p(x)# is a polynomial, and #(x-a)# is a factor of that polynomial

Then, #p(a)=(x-a)q(a)+0#

#q(a)# is the quotient and #0# the remainder

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

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

#=16-20+8-4=24-24=0#

The remainder is #=0#