Divide Using Long Polynomial Division (2x^4+7x^3+2x^2+5x-4)/(x^2+3x-1)

$\frac{2 x^{4} + 7 x^{3} + 2 x^{2} + 5 x - 4}{x^{2} + 3 x - 1}$

Step 1

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

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

Step 2

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

$2 x^{2}$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + 6 x^{3} - 2 x^{2}$

$2 x^{2}$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

Step 5

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

$2 x^{2}$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

Step 6

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

$2 x^{2}$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

Step 7

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

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$x$

Step 9

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

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$x$

Step 10

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

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$x$

$x^{2}$

$6 x$

Step 11

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

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$x$

$x^{2}$

$6 x$

$4$

Step 12

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

$2 x^{2}$

$x$

$1$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$x$

$x^{2}$

$6 x$

$4$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x$

$1$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$x$

$x^{2}$

$6 x$

$4$

$x^{2}$

$3 x$

$1$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 3 x - 1$

$2 x^{2}$

$x$

$1$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$x$

$x^{2}$

$6 x$

$4$

$x^{2}$

$3 x$

$1$

Step 15

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

$2 x^{2}$

$x$

$1$

$x^{2}$

$3 x$

$1$

$2 x^{4}$

$7 x^{3}$

$2 x^{2}$

$5 x$

$4$

$2 x^{4}$

$6 x^{3}$

$2 x^{2}$

$x^{3}$

$4 x^{2}$

$5 x$

$x^{3}$

$3 x^{2}$

$x$

$x^{2}$

$6 x$

$4$

$x^{2}$

$3 x$

$1$

$3 x$

$3$

Step 16

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

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

Divide Using Long Polynomial Division (2x^4+7x^3+2x^2+5x-4)/(x^2+3x-1)

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