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

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Algebra

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

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

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

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

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

$x^{4}$

$0$

$x^{2}$

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

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

$x^{4}$

$0$

$x^{2}$

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

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

$x^{4}$

$0$

$x^{2}$

$5 x^{3}$

$3 x^{2}$

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

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

$x^{4}$

$0$

$x^{2}$

$5 x^{3}$

$3 x^{2}$

$5 x$

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$5 x$

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

$x^{4}$

$0$

$x^{2}$

$5 x^{3}$

$3 x^{2}$

$5 x$

Multiply the new quotient term by the divisor.

$x^{2}$

$5 x$

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

$x^{4}$

$0$

$x^{2}$

$5 x^{3}$

$3 x^{2}$

$5 x$

$5 x^{3}$

$0$

$5 x$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 0 + 5 x$

$x^{2}$

$5 x$

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

$x^{4}$

$0$

$x^{2}$

$5 x^{3}$

$3 x^{2}$

$5 x$

$5 x^{3}$

$0$

$5 x$

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

$x^{2}$

$5 x$

$x^{2}$

$0 x$

$1$

$x^{4}$

$5 x^{3}$

$2 x^{2}$

$5 x$

$3$

$x^{4}$

$0$

$x^{2}$

$5 x^{3}$

$3 x^{2}$

$5 x$

$5 x^{3}$

$0$

$5 x$

$3 x^{2}$

$0$

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 5 x - 3$

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