Divide (x^3+x^2+x+2)÷(x^2-1)

$(x^{3} + x^{2} + x + 2) \div (x^{2} - 1)$

Step 1

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

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

Step 2

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{3}$

$0$

$x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 - x$

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{3}$

$0$

$x$

Step 5

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{3}$

$0$

$x$

$x^{2}$

$2 x$

Step 6

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

$x$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{3}$

$0$

$x$

$x^{2}$

$2 x$

$2$

Step 7

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

$x$

$1$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{3}$

$0$

$x$

$x^{2}$

$2 x$

$2$

Step 8

Multiply the new quotient term by the divisor.

$x$

$1$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{3}$

$0$

$x$

$x^{2}$

$2 x$

$2$

$x^{2}$

$0$

$1$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + 0 - 1$

$x$

$1$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{3}$

$0$

$x$

$x^{2}$

$2 x$

$2$

$x^{2}$

$0$

$1$

Step 10

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

$x$

$1$

$x^{2}$

$0 x$

$1$

$x^{3}$

$x^{2}$

$x$

$2$

$x^{3}$

$0$

$x$

$x^{2}$

$2 x$

$2$

$x^{2}$

$0$

$1$

$2 x$

$3$

Step 11

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

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

Divide (x^3+x^2+x+2)÷(x^2-1)

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