Factor x^4+2x^3+2x^2-2x-3

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

Regroup terms.

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

Factor $2 x^{2}$ out of $2 x^{3} + 2 x^{2}$ .

Factor $2 x^{2}$ out of $2 x^{3}$ .

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

Factor $2 x^{2}$ out of $2 x^{2}$ .

$2 x^{2} (x) + 2 x^{2} (1) + x^{4} - 2 x - 3$

Factor $2 x^{2}$ out of $2 x^{2} (x) + 2 x^{2} (1)$ .

$2 x^{2} (x + 1) + x^{4} - 2 x - 3$

Factor $x^{4} - 2 x - 3$ 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.

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

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

$\pm 1, \pm 3$

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} - 2 \cdot - 1 - 3$

Raise $- 1$ to the power of $4$ .

$1 - 2 \cdot - 1 - 3$

Multiply $- 2$ by $- 1$ .

$1 + 2 - 3$

Add $1$ and $2$ .

$3 - 3$

Subtract $3$ from $3$ .

$0$

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

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

Divide $x^{4} - 2 x - 3$ by $x + 1$ .

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

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

$x^{2}$

$x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

$x^{2}$

$x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

$x^{2}$

$x$

$3 x$

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

$x^{2}$

$x$

$3 x$

$3$

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

$x^{3}$

$x^{2}$

$x$

$3$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

$x^{2}$

$x$

$3 x$

$3$

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x - 3$

$x^{3}$

$x^{2}$

$x$

$3$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

$x^{2}$

$x$

$3 x$

$3$

$3 x$

$3$

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

$x^{3}$

$x^{2}$

$x$

$3$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$2 x$

$3$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$2 x$

$x^{2}$

$x$

$3 x$

$3$

$3 x$

$3$

$0$

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

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

Write $x^{4} - 2 x - 3$ as a set of factors.

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

Factor $x + 1$ out of $2 x^{2} (x + 1) + (x + 1) (x^{3} - x^{2} + x - 3)$ .

Factor $x + 1$ out of $2 x^{2} (x + 1)$ .

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

Factor $x + 1$ out of $(x + 1) (2 x^{2}) + (x + 1) (x^{3} - x^{2} + x - 3)$ .

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

Subtract $x^{2}$ from $2 x^{2}$ .

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

Factor.

Factor $x^{3} + x^{2} + x - 3$ 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.

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

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

$\pm 1, \pm 3$

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

Substitute $1$ into the polynomial.

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

Raise $1$ to the power of $3$ .

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

Raise $1$ to the power of $2$ .

$1 + 1 + 1 - 3$

Add $1$ and $1$ .

$2 + 1 - 3$

Add $2$ and $1$ .

$3 - 3$

Subtract $3$ from $3$ .

$0$

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

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

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

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

$x$

$1$

$x^{3}$

$x^{2}$

$x$

$3$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

The expression needs to be subtracted from the dividend, so change all the signs in $3 x - 3$

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

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

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

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

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

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

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

Remove unnecessary parentheses.

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

Factor x^4+2x^3+2x^2-2x-3

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