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

$\dfrac{x^3+10x^2+13x+39}{x^2+2x+1}$

$x^3=x\cdot x^2$, 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

$\dfrac{x^3+10x^2+13x+39}{x^2+2x+1}=x+8+\dfrac{-4x+31}{x^2+2x+1}$