Divide Using Long Polynomial Division Use the long division method to find the result when 6x^3-5x^2+22x-7 is divided by 3x-1

Use the long division method to find the result when $6 x^{3} - 5 x^{2} + 22 x - 7$ is divided by $3 x - 1$

Write the problem as a mathematical expression.

Use the long division method to find the result when $\frac{6 x^{3} - 5 x^{2} + 22 x - 7}{3 x - 1}$

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

$3 x$

$1$

$6 x^{3}$

$5 x^{2}$

$22 x$

$7$

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

					$2 x^{2}$
	$3 x$	-	$1$		$6 x^{3}$	-	$5 x^{2}$	+	$22 x$	-	$7$

Multiply the new quotient term by the divisor.

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

Divide the highest order term in the dividend $21 x$ by the highest order term in divisor $3 x$ .

Multiply the new quotient term by the divisor.

The expression needs to be subtracted from the dividend, so change all the signs in $21 x - 7$

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

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

$2 x^{2} - x + 7$