Divide 2x^2 - 1 by 2x + 1. Give the quotient and the remainder.

Step-by-step solution

1. 1

   Insert the missing term. Long division lines up by degree, so write the dividend with every power present: $2x^2-1=2x^2+0x-1.$ Skipping this is the single most common source of sign errors.

2. 2

   Divide the leading terms. $\dfrac{2x^2}{2x}=x$, so the first quotient term is $x$. Multiply back: $x(2x+1)=2x^2+x$, and subtract: $(2x^2+0x-1)-(2x^2+x)=-x-1.$

3. 3

   Repeat on the new remainder. $\dfrac{-x}{2x}=-\dfrac12$. Because the divisor is non-monic, the quotient picks up a fraction — that is expected, not a mistake. Multiply back: $-\tfrac12(2x+1)=-x-\tfrac12$.

4. 4

   Subtract again and stop. $(-x-1)-\left(-x-\frac12\right)=-\frac12.$ The degree of $-\tfrac12$ is below the degree of $2x+1$, so the division terminates.

5. 5

   Check with the division identity. $\left(x-\frac12\right)(2x+1)+\left(-\frac12\right)=2x^2+x-x-\frac12-\frac12=2x^2-1,$ which is the original dividend.