1 Answer

Tomas Kirda · Answer 1 · 2020-12-25T10:52:43+0000

$4x^5=x^2\cdot4x^3$ , and
4x^3(x^2-2x)=4x^5-8x^4 . Subtract this from
4x^5-3x^3+2x-1 to get a remainder of

(4x^5-3x^3+2x-1)-(4x^5-8x^4)=8x^4-3x^3+2x-1

$8x^4=x^2\cdot8x^2$ , and
8x^2(x^2-2x)=8x^4-16x^3 . Subtract this from the previous remainder to get a new remainder of

(8x^4-3x^3+2x-1)-(8x^4-16x^3)=13x^3+2x-1

$13x^3=x^2\cdot13x$ , and
13x(x^2-2x)=13x^3-26x^2 . Subtract this from the previous remainder to get a new remainder of

(13x^3+2x-1)-(13x^3-26x^2)=26x^2+2x-1

$26x^2=x^2\cdot26$ , and
26(x^2-2x)=26x^2-52x . Subtract this from the previous remainder to get a new remainder of

(26x^2+2x-1)-(26x^2-52x)=54x-1

x^2 does not divide
54x , so we're done, and we've found that

(4x^5-3x^3+2x-1)/(x^2-2x)=4x^3+(8x^4-3x^3+2x-1)/(x^2-2x)

(4x^5-3x^3+2x-1)/(x^2-2x)=4x^3+8x^2+(13x^3+2x-1)/(x^2-2x)

(4x^5-3x^3+2x-1)/(x^2-2x)=4x^3+8x^2+13x+(26x^2+2x-1)/(x^2-2x)

$(4x^5-3x^3+2x-1)/(x^2-2x)=\boxed{4x^3+8x^2+13x+26+(54x-1)/(x^2-2x)}$

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