How do you divide #(x^4-5x^3+2x^2-4) / (x^3-4x) # using polynomial long division?

1 Answer
Sep 5, 2016

#x-5+(6x^2-20x-4)/(x^3-4x)#

Explanation:

Incorporating place keepers such as #0x# so that things line up properly.

#" "x^4-5x^3+2x^2+0x-4#
#color(magenta)(x)(x^3-4x) ->" "ul( x^4+0x^3-4x^2) larr" Subtract"#
#" "0 -5x^3+6x^2+0x-4#
#color(magenta)(-5)(x^3-4x)->" "color(white)()ul(-5x^3+0x^2+20x ) larr" Subtract"#
#" "color(blue)(0 +6x^2-20x-4) larr" Remainder"#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Highest order of #6x^2-20x-4 # is 2 from #6x^2#

Highest order of the divisor #x^3-4x#

As the order of the divisor is now the highest we stop and we have to express the remainder in fraction form with the divisor as the denominator giving:

#(x^4-5x^3+2x^2-4) -: (x^3-4x) = color(magenta)(x-5 +(color(blue)(6x^2-20x-4))/(x^3-4x) #