We can do it by algebraic long division:
#"dividend"/"divisor" = "quotient"#
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Write the dividend in the 'box' making sure that the indices are in descending powers of x. Make spaces for any missing terms.
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Divide the first term in the divisor into the term in the dividend with the highest index. Write the answer at the top,
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Multiply by BOTH terms of the divisor at the side
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Subtract
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Bring down the next term
Repeat steps 2 to 5
#color(white)(xxxxxx.xxxxxx)color(red)(x^2)" " color(blue)(+xy)" "+color(lime)(y^2) " rem "2y^3 #
#color(white)(xxx)x-y |bar( x^3 +0x^2y+0xy^2 +y^3" "larrx^3divx =color(red)(x^2#
#color(white)(xxxxx)-(ul(color(red)(x^3-x^2y)))" "larr# subtract
#color(white)(xxxxxxxxxx) +x^2y" "larrx^2y div x = color(blue)(xy)#
#color(white)(xxxxxx.x.)-(ul(color(blue)(x^2y-xy^2)))color(white)(x)darr" "larr# subtract
#color(white)(xxxxxxxxxxx.x.xxx)xy^2 -y^3 " "larrxy^2divx = color(lime)(y^2)#
#color(white)(xxxxxxxx..x.xxx)ul(-color(lime)((xy^2 +y^3)) " "larr# subtract
#color(white)(xxxxxxxxxxxxxxxxxxxx)2y^3larr" "# remainder
#(x^3 + y^3) div(x-y) = x^2 +xy +y^2 " rem " 2y^3#
