Question

How do you use long division to divide `(x^4-2x^3-4x^2+2x+3) -: (x^2+2x+1)`?

Answer 1

`(x^4-2x^3-4x^2+2x+3) -: (x^2+2x+1) = x^2-4x+3`, with no remainder.

Explanation:

Set up the long division like this:

`color(magenta)(x^2)+2x+1bar(|"  "color(magenta)(x^4)-2x^3-4x^2+2x+3)`

Divide `color(magenta)(x^4/x^2)`, giving `color(red)(x^2)`; put this quotient above the `x^4`:

`color(white)(x^2+2x+1bar(|"  "color(red)(x^2)`
`x^2+2x+1bar(|"  "x^4-2x^3-4x^2+2x+3)`

Multiply `color(red)(x^2) xx (x^2+2x+1)`, giving `color(blue)(x^4+2x^3+x^2)`; put this product below the `x^4-2x^3-4x^2`:

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2)`
`x^2+2x+1bar(|"  "x^4-2x^3-4x^2+2x+3)`
`color(white)(x^2+2x+1bar(|"  "color(blue)(x^4+2x^3+color(white)(1)x^2))`

Subtract `(x^4-2x^3-4x^2)-(color(blue)(x^4+2x^3+x^2))`, giving `color(orange)(–4x^3-5x^2)`; draw a line under `color(blue)(x^4+2x^3+x^2)` and write this difference below the line:

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2)`
`x^2+2x+1bar(|"  "x^4-2x^3-4x^2+2x+3)`
`color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))`
`color(white)(x^2+2x+1|)bar("  "color(white)(x^4"  ")color(orange)(-4x^3-5x^2)"          ")`

Copy the `color(green)(2x)` from the dividend down below this line:

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2)`
`x^2+2x+1bar(|"  "x^4-2x^3-4x^2+color(green)(2x)+3)`
`color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))`
`color(white)(x^2+2x+1|)bar("  "color(white)(x^4)-4x^3-5x^2" "color(green)(+2x))`

Repeat this process twice, dividing the latest leading term below your line by the leading `x^2` from the divisor:

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2-color(red)(4x))`
`color(magenta)(x^2)+2x+1bar(|"  "x^4-2x^3-4x^2+2x+3)`
`color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))`
`color(white)(x^2+2x+1|)bar("  "color(white)(x^4)-color(magenta)(4x^3)-5x^2+2x)`

...

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2-color(red)(4x))`
`color(red)(x^2+2x+1)bar(|"  "x^4-2x^3-4x^2+2x+3)`
`color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))`
`color(white)(x^2+2x+1|)bar("  "color(white)(x^4)-4x^3-5x^2+2x)`
`color(white)(x^2+2x+1|bar("     "color(blue)(-4x^3-8x^2-4x)))`

...

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2-4x)`
`x^2+2x+1bar(|"  "x^4-2x^3-4x^2+2x+color(green)(3))`
`color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))`
`color(white)(x^2+2x+1|)bar("  "color(white)(x^4)-4x^3-5x^2+2x)`
`color(white)(x^2+2x+1|bar("     "color(black)(-4x^3-8x^2-4x)))`
`color(white)(x^2+2x+1|"        ")bar(color(white)(-4x^3+)color(orange)(3x^2+6x)+color(green)(3))`

...

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2-4x"  "+color(red)3)`
`color(magenta)(x^2)+2x+1bar(|"  "x^4-2x^3-4x^2+2x+3)`
`color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))`
`color(white)(x^2+2x+1|)bar("  "color(white)(x^4)-4x^3-5x^2+2x)`
`color(white)(x^2+2x+1|bar("     "color(black)(-4x^3-8x^2-4x)))`
`color(white)(x^2+2x+1|"        ")bar(color(white)(-4x^3+)color(magenta)(3x^2)+6x+3)`

...

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2-4x"  "+color(red)(3))`
`color(red)(x^2+2x+1)bar(|"  "x^4-2x^3-4x^2+2x+3)`
`color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))`
`color(white)(x^2+2x+1|)bar("  "color(white)(x^4)-4x^3-5x^2+2x)`
`color(white)(x^2+2x+1|bar("     "color(black)(-4x^3-8x^2-4x)))`
`color(white)(x^2+2x+1|"        ")bar(color(white)(-4x^3+)3x^2+6x+3)`
`color(white)(x^2+2x+1|"        "bar(color(blue)("            "3x^2+6x+3)`

...

`color(white)(x^2+2x+1bar(|"  "color(black)(x^2-4x"  "+3)`
`x^2+2x+1bar(|"  "x^4-2x^3-4x^2+2x+3)`
`color(white)(x^2+2x+1bar(|"  "color(black)(x^4+2x^3+color(white)(1)x^2))`
`color(white)(x^2+2x+1|)bar("  "color(white)(x^4)-4x^3-5x^2+2x)`
`color(white)(x^2+2x+1|bar("     "color(black)(-4x^3-8x^2-4x)))`
`color(white)(x^2+2x+1|"        ")bar(color(white)(-4x^3+)3x^2+6x+3)`
`color(white)(x^2+2x+1|"        "bar(color(black)("            "3x^2+6x+3)`
`color(white)(x^2+2x+1|"                       ")bar(color(white)(3x^2+6x+color(orange)0)`