Find the Remainder (12x^3+20x^2+23x+8)/(2x+1)

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of ${\displaystyle 0}$.

Divide the highest order term in the dividend ${\displaystyle 12x^3}$ by the highest order term in divisor ${\displaystyle 2x}$.

Multiply the new quotient term by the divisor.

The expression needs to be subtracted from the dividend, so change all the signs in ${\displaystyle 12x^3+6x^2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

Pull the next terms from the original dividend down into the current dividend.

Divide the highest order term in the dividend ${\displaystyle 14x^2}$ by the highest order term in divisor ${\displaystyle 2x}$.

Multiply the new quotient term by the divisor.

The expression needs to be subtracted from the dividend, so change all the signs in ${\displaystyle 14x^2+7x}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

Pull the next terms from the original dividend down into the current dividend.

Divide the highest order term in the dividend ${\displaystyle 16x}$ by the highest order term in divisor ${\displaystyle 2x}$.

Multiply the new quotient term by the divisor.

The expression needs to be subtracted from the dividend, so change all the signs in ${\displaystyle 16x+8}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

Since the remander is ${\displaystyle 0}$, the final answer is the quotient.