Finite Math Examples

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

Step 1

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

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

Step 2

Divide the highest order term in the dividend $5 x^{4}$ by the highest order term in divisor $x^{2}$ .

$5 x^{2}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

Step 3

Multiply the new quotient term by the divisor.

$5 x^{2}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

$5 x^{4}$

$0$

$10 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} + 0 + 10 x^{2}$

$5 x^{2}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

$5 x^{4}$

$0$

$10 x^{2}$

Step 5

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

$5 x^{2}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

$5 x^{4}$

$0$

$10 x^{2}$

$13 x^{2}$

Step 6

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

$5 x^{2}$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

$5 x^{4}$

$0$

$10 x^{2}$

$13 x^{2}$

$x$

$1$

Step 7

Divide the highest order term in the dividend $- 13 x^{2}$ by the highest order term in divisor $x^{2}$ .

$5 x^{2}$

$0 x$

$13$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

$5 x^{4}$

$0$

$10 x^{2}$

$13 x^{2}$

$x$

$1$

Step 8

Multiply the new quotient term by the divisor.

						$5 x^{2}$	+	$0 x$	-	$13$
$x^{2}$	+	$0 x$	+	$2$		$5 x^{4}$	+	$0 x^{3}$	-	$3 x^{2}$	+	$x$	+	$1$
					-	$5 x^{4}$	-	$0$	-	$10 x^{2}$
									-	$13 x^{2}$	+	$x$	+	$1$
									-	$13 x^{2}$	+	$0$	-	$26$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x^{2} + 0 - 26$

$5 x^{2}$

$0 x$

$13$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

$5 x^{4}$

$0$

$10 x^{2}$

$13 x^{2}$

$x$

$1$

$13 x^{2}$

$0$

$26$

Step 10

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

$5 x^{2}$

$0 x$

$13$

$x^{2}$

$0 x$

$2$

$5 x^{4}$

$0 x^{3}$

$3 x^{2}$

$x$

$1$

$5 x^{4}$

$0$

$10 x^{2}$

$13 x^{2}$

$x$

$1$

$13 x^{2}$

$0$

$26$

$x$

$27$

Step 11

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

$5 x^{2} - 13 + \frac{x + 27}{x^{2} + 2}$