Divide Using Long Polynomial Division (2x^4+3x^3+7x^2+7x+6) by (2x+1)

$(2 x^{4} + 3 x^{3} + 7 x^{2} + 7 x + 6)$ by $(2 x + 1)$

Write the problem as a mathematical expression.

$\frac{(2 x^{4} + 3 x^{3} + 7 x^{2} + 7 x + 6)}{(2 x + 1)}$

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

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

Divide the highest order term in the dividend $2 x^{4}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + x^{3}$

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

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

$x^{3}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

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

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

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

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

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

$x^{3}$

$x^{2}$

$3 x$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $2 x$ .

$x^{3}$

$x^{2}$

$3 x$

$2$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$3 x$

$2$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

$4 x$

$2$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 2$

$x^{3}$

$x^{2}$

$3 x$

$2$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

$4 x$

$2$

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

$x^{3}$

$x^{2}$

$3 x$

$2$

$2 x$

$1$

$2 x^{4}$

$3 x^{3}$

$7 x^{2}$

$7 x$

$6$

$2 x^{4}$

$x^{3}$

$2 x^{3}$

$7 x^{2}$

$2 x^{3}$

$x^{2}$

$6 x^{2}$

$7 x$

$6 x^{2}$

$3 x$

$4 x$

$6$

$4 x$

$2$

$4$

The final answer is the quotient plus the remainder over the divisor.

$x^{3} + x^{2} + 3 x + 2 + \frac{4}{2 x + 1}$

Divide Using Long Polynomial Division (2x^4+3x^3+7x^2+7x+6) by (2x+1)

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