Divide (6x^3-1x^2+1)÷(2x+1)

$(6 x^{3} - 1 x^{2} + 1) \div (2 x + 1)$

Step 1

Rewrite the division as a fraction.

$\frac{6 x^{3} - 1 x^{2} + 1}{2 x + 1}$

Step 2

Simplify the numerator.

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{6 x^{3} - x^{2} + 1}{2 x + 1}$

Factor $6 x^{3} - x^{2} + 1$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1 q = \pm 1, \pm 6, \pm 2, \pm 3$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}$

Substitute $- 0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.5$ is a root of the polynomial.

Substitute $- 0.5$ into the polynomial.

$6 {(- 0.5)}^{3} - {(- 0.5)}^{2} + 1$

Raise $- 0.5$ to the power of $3$ .

$6 \cdot - 0.125 - {(- 0.5)}^{2} + 1$

Multiply $6$ by $- 0.125$ .

$- 0.75 - {(- 0.5)}^{2} + 1$

Raise $- 0.5$ to the power of $2$ .

$- 0.75 - 1 \cdot 0.25 + 1$

Multiply $- 1$ by $0.25$ .

$- 0.75 - 0.25 + 1$

Subtract $0.25$ from $- 0.75$ .

$- 1 + 1$

Add $- 1$ and $1$ .

$0$

Since $- 0.5$ is a known root, divide the polynomial by $2 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{6 x^{3} - x^{2} + 1}{2 x + 1}$

Divide $6 x^{3} - x^{2} + 1$ by $2 x + 1$ .

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

$2 x$

$1$

$6 x^{3}$

$x^{2}$

$0 x$

$1$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $2 x$ .

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

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $2 x + 1$

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

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

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

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

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

Write $6 x^{3} - x^{2} + 1$ as a set of factors.

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

Step 3

Cancel the common factor of $2 x + 1$ .

Cancel the common factor.

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

Divide $3 x^{2} - 2 x + 1$ by $1$ .

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

Divide (6x^3-1x^2+1)÷(2x+1)

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