Divide (6x^3+2x^2+3x-4)÷(x^2+3x+4)

$(6 x^{3} + 2 x^{2} + 3 x - 4) \div (x^{2} + 3 x + 4)$

Step 1

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

$x^{2}$

$3 x$

$4$

$6 x^{3}$

$2 x^{2}$

$3 x$

$4$

Step 2

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $x^{2}$ .

$6 x$

$x^{2}$

$3 x$

$4$

$6 x^{3}$

$2 x^{2}$

$3 x$

$4$

Step 3

Multiply the new quotient term by the divisor.

$6 x$

$x^{2}$

$3 x$

$4$

$6 x^{3}$

$2 x^{2}$

$3 x$

$4$

$6 x^{3}$

$18 x^{2}$

$24 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} + 18 x^{2} + 24 x$

$6 x$

$x^{2}$

$3 x$

$4$

$6 x^{3}$

$2 x^{2}$

$3 x$

$4$

$6 x^{3}$

$18 x^{2}$

$24 x$

Step 5

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

$6 x$

$x^{2}$

$3 x$

$4$

$6 x^{3}$

$2 x^{2}$

$3 x$

$4$

$6 x^{3}$

$18 x^{2}$

$24 x$

$16 x^{2}$

$21 x$

Step 6

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

						$6 x$
$x^{2}$	+	$3 x$	+	$4$		$6 x^{3}$	+	$2 x^{2}$	+	$3 x$	-	$4$
					-	$6 x^{3}$	-	$18 x^{2}$	-	$24 x$
							-	$16 x^{2}$	-	$21 x$	-	$4$

Step 7

Divide the highest order term in the dividend $- 16 x^{2}$ by the highest order term in divisor $x^{2}$ .

						$6 x$	-	$16$
$x^{2}$	+	$3 x$	+	$4$		$6 x^{3}$	+	$2 x^{2}$	+	$3 x$	-	$4$
					-	$6 x^{3}$	-	$18 x^{2}$	-	$24 x$
							-	$16 x^{2}$	-	$21 x$	-	$4$

Step 8

Multiply the new quotient term by the divisor.

						$6 x$	-	$16$
$x^{2}$	+	$3 x$	+	$4$		$6 x^{3}$	+	$2 x^{2}$	+	$3 x$	-	$4$
					-	$6 x^{3}$	-	$18 x^{2}$	-	$24 x$
							-	$16 x^{2}$	-	$21 x$	-	$4$
							-	$16 x^{2}$	-	$48 x$	-	$64$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x^{2} - 48 x - 64$

$6 x$

$16$

$x^{2}$

$3 x$

$4$

$6 x^{3}$

$2 x^{2}$

$3 x$

$4$

$6 x^{3}$

$18 x^{2}$

$24 x$

$16 x^{2}$

$21 x$

$4$

$16 x^{2}$

$48 x$

$64$

Step 10

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

$6 x$

$16$

$x^{2}$

$3 x$

$4$

$6 x^{3}$

$2 x^{2}$

$3 x$

$4$

$6 x^{3}$

$18 x^{2}$

$24 x$

$16 x^{2}$

$21 x$

$4$

$16 x^{2}$

$48 x$

$64$

$27 x$

$60$

Step 11

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

$6 x - 16 + \frac{27 x + 60}{x^{2} + 3 x + 4}$

Divide (6x^3+2x^2+3x-4)÷(x^2+3x+4)

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