(2x^4+5x^3-3x+1)/(x^2-x+2)
We can use the polynomial long division method, begin by writing:
x^2-x+2|overline(2x^4+5x^3-3x+1)
The first question we have to ask is how many times does x^2 go into 2x^4. The answer is obviously 2x^2. We put this factor on top, so the next line of our working will look like:
color(white)(...................)2x^2
x^2-x+2|overline(2x^4+5x^3-3x+1)
We now have to multiply the divisor x^2-x+2 by 2x^2 to get
2x^4-2x^3+4x^2 and subtract this from the dividend, the next line of working will look like:
color(white)(...................)2x^2
x^2-x+2|overline(2x^4+5x^3-0x^2-3x+1)
color(white)(...............)ul(-(2x^4-2x^3+4x^2))
Notice in the dividend their is no x^2 so I have added a 0x^2 term in place of where the x^2 term should be to make the working clearer. We do the subtraction to get:
color(white)(...................)2x^2
x^2-x+2|overline(2x^4+5x^3-0x^2-3x+1)
color(white)(...............)ul(-(2x^4-2x^3+4x^2+0x+0))
color(white)(......................)0+7x^3-4x^2-3x+1
We now repeat the process, treating: 7x^3-4x^2+3x+1 as the "new dividend." Keep repeating until the degree of the dividend is less than the degree of the divisor. The working, when written out in full will look like:
color(white)(...................)2x^2+7x+3
x^2-x+2|overline(2x^4+5x^3-0x^2-3x+1)
color(white)(...............)ul(-(2x^4-2x^3+4x^2+0x+0))
color(white)(......................)0+7x^3-4x^2-3x+1
color(white)(.......................)ul(-(7x^3-7x^2+14x+0))
color(white)(................................)0+3x^2-17x+1
color(white)(....................................)ul(-(3x^2-3x+6))
color(white)(.............................................)-14x-5
So the result is:
2x^2+7x+3 with a remainder of -14x-5. We can also use this to rewrite the original fraction as:
(2x^4+5x^3-3x+1)/(x^2-x+2)=2x^2+7x+3+(-14x-5)/(x^2-x+2)
