Find the Quotient (3x^4-x^2)/(x^3-x^2+1)

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

Step 1

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

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x^{3}$ .

$3 x$

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$3 x$

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$3 x^{4}$

$3 x^{3}$

$0$

$3 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} - 3 x^{3} + 0 + 3 x$

$3 x$

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$3 x^{4}$

$3 x^{3}$

$0$

$3 x$

Step 5

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

$3 x$

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$3 x^{4}$

$3 x^{3}$

$0$

$3 x$

$3 x^{3}$

$x^{2}$

$3 x$

Step 6

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

$3 x$

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$3 x^{4}$

$3 x^{3}$

$0$

$3 x$

$3 x^{3}$

$x^{2}$

$3 x$

$0$

Step 7

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

$3 x$

$3$

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$3 x^{4}$

$3 x^{3}$

$0$

$3 x$

$3 x^{3}$

$x^{2}$

$3 x$

$0$

Step 8

Multiply the new quotient term by the divisor.

$3 x$

$3$

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$3 x^{4}$

$3 x^{3}$

$0$

$3 x$

$3 x^{3}$

$x^{2}$

$3 x$

$0$

$3 x^{3}$

$3 x^{2}$

$0$

$3$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 3 x^{2} + 0 + 3$

$3 x$

$3$

$x^{3}$

$x^{2}$

$0 x$

$1$

$3 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$0$

$3 x^{4}$

$3 x^{3}$

$0$

$3 x$

$3 x^{3}$

$x^{2}$

$3 x$

$0$

$3 x^{3}$

$3 x^{2}$

$0$

$3$

Step 10

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

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

Step 11

The quotient is $3 x + 3$ .

$3 x + 3$

Find the Quotient (3x^4-x^2)/(x^3-x^2+1)

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