Algebra Examples

$\frac{x^{4} + 6 x^{2} - 5}{x^{4}}$

Step 1

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

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$5$

Step 2

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

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$5$

Step 3

Multiply the new quotient term by the divisor.

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$0$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$5$

$x^{4}$

$0$

Step 4

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

										$1$
$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$		$x^{4}$	+	$0 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$5$
									-	$x^{4}$	-	$0$	-	$0$	-	$0$	-	$0$

Step 5

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

										$1$
$x^{4}$	+	$0 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$		$x^{4}$	+	$0 x^{3}$	+	$6 x^{2}$	+	$0 x$	-	$5$
									-	$x^{4}$	-	$0$	-	$0$	-	$0$	-	$0$
													+	$6 x^{2}$	+	$0$	-	$5$

Step 6

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

$1 + \frac{6 x^{2} - 5}{x^{4}}$