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

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

Step 1

Regroup terms.

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

Step 2

Factor $x^{2}$ out of $x^{4} - x^{2}$ .

Factor $x^{2}$ out of $x^{4}$ .

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

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

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

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 1$ .

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

Step 3

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

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

Step 4

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Remove unnecessary parentheses.

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

Step 5

Factor $3 x^{3} + x + 4$ 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.

$p = \pm 1, \pm 4, \pm 2 q = \pm 1, \pm 3$

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

$\pm 1, \pm 0.\overline{3}, \pm 4, \pm 1.\overline{3}, \pm 2, \pm 0.\overline{6}$

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.

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

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

$3 \cdot - 1 - 1 + 4$

Multiply $3$ by $- 1$ .

$- 3 - 1 + 4$

Subtract $1$ from $- 3$ .

$- 4 + 4$

Add $- 4$ and $4$ .

$0$

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.

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

Divide $3 x^{3} + x + 4$ 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 $0$ .

$x$

$1$

$3 x^{3}$

$0 x^{2}$

$x$

$4$

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Multiply the new quotient term by the divisor.

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

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

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

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

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

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

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

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

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

Step 6

Factor $x + 1$ out of $x^{2} (x + 1) (x - 1) + (x + 1) (3 x^{2} - 3 x + 4)$ .

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

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

Factor $x + 1$ out of $(x + 1) (x^{2} (x - 1)) + (x + 1) (3 x^{2} - 3 x + 4)$ .

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

Step 7

Apply the distributive property.

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

Step 8

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

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

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $2$ and $1$ .

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

Step 9

Move $- 1$ to the left of $x^{2}$ .

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

Step 10

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

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

Step 11

Add $- x^{2}$ and $3 x^{2}$ .

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

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

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