Question

Rewrite the expression x^3+10x^2+13x+39/x^2+2x+1 in the form of q(x)+r(x)/b(x)

Answer 1

`(x^3+10x^2+13x+39)/(x^2+2x+1)`

`x^3=x\cdot x^2`, and
`x(x^2+2x+1)=x^3+2x^2+x`. Subtracting this from the numerator gives a remainder of

`(x^3+10x^2+13x+39)-(x^3+2x^2+x)=8x^2+12x+39`

`8x^2=8\cdot x^2`, and
`8(x^2+2x+1)=8x^2+16x+8`. Subtracting this from the previous remainder gives a new remainder of

`(8x^2+12x+39)-(8x^2+16x+8)=-4x+31`

`-84x` is not a multiple of
`x^2`, so we're done. Then

`(x^3+10x^2+13x+39)/(x^2+2x+1)=x+8+(-4x+31)/(x^2+2x+1)`