+118651 What is the quotient and the remainder of (4x^3 - 2x^2 + 1) / ( -x + 2 )?'
\(\frac{(-4x^3 + 2x^2 - 1) }{( x - 2 )}\\ -4*2^3+2*2^2-1=-4*8+8-1=-25\\ \mbox{so using remainder theorem the remainder will be -25}\\ =\frac{(-4x^3 + 2x^2 +24)-25 }{( x - 2 )}\\ =\frac{2[-2x^3 + x^2 +12]-25 }{( x - 2 )}\\ =\frac{2[(x-2)(-2x^2-3x-6]-25 }{( x - 2 )}\\ =\frac{2[(x-2)(-2x^2-3x-6] }{( x - 2 )}+\frac{-25 }{( x - 2 )}\\ =2[-2x^2-3x-6] +\frac{-25 }{( x - 2 )}\\ \mbox{so the answer is}\\ -4x^2-6x-12\;\;remainder \;\;-25 \)
Mind you, this was a very weird way to do it - normally you would just use algebraic division and you'd get to the answer more simply :))
