Divide Using Long Polynomial Division (2x^3+3x^2+4x+5)÷(2x^2+3x+4)

$(2 x^{3} + 3 x^{2} + 4 x + 5) \div (2 x^{2} + 3 x + 4)$

Step 1

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

$2 x^{2}$

$3 x$

$4$

$2 x^{3}$

$3 x^{2}$

$4 x$

$5$

Step 2

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

$x$

$2 x^{2}$

$3 x$

$4$

$2 x^{3}$

$3 x^{2}$

$4 x$

$5$

Step 3

Multiply the new quotient term by the divisor.

$x$

$2 x^{2}$

$3 x$

$4$

$2 x^{3}$

$3 x^{2}$

$4 x$

$5$

$2 x^{3}$

$3 x^{2}$

$4 x$

Step 4

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

$x$

$2 x^{2}$

$3 x$

$4$

$2 x^{3}$

$3 x^{2}$

$4 x$

$5$

$2 x^{3}$

$3 x^{2}$

$4 x$

Step 5

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

$x$

$2 x^{2}$

$3 x$

$4$

$2 x^{3}$

$3 x^{2}$

$4 x$

$5$

$2 x^{3}$

$3 x^{2}$

$4 x$

$0$

Step 6

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

$x$

$2 x^{2}$

$3 x$

$4$

$2 x^{3}$

$3 x^{2}$

$4 x$

$5$

$2 x^{3}$

$3 x^{2}$

$4 x$

$0$

$5$

Step 7

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

$x + \frac{5}{2 x^{2} + 3 x + 4}$

Divide Using Long Polynomial Division (2x^3+3x^2+4x+5)÷(2x^2+3x+4)

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