Algebra Examples

$x^{4} - 5 x^{3} + 5 x^{2} + 5 x - 6$

Step 1

Regroup terms.

$- 5 x^{3} + 5 x^{2} + x^{4} + 5 x - 6$

Step 2

Factor $- 5 x^{2}$ out of $- 5 x^{3} + 5 x^{2}$ .

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Step 2.1

Factor $- 5 x^{2}$ out of $- 5 x^{3}$ .

$- 5 x^{2} (x) + 5 x^{2} + x^{4} + 5 x - 6$

Step 2.2

Factor $- 5 x^{2}$ out of $5 x^{2}$ .

$- 5 x^{2} (x) - 5 x^{2} (- 1) + x^{4} + 5 x - 6$

Step 2.3

Factor $- 5 x^{2}$ out of $- 5 x^{2} (x) - 5 x^{2} (- 1)$ .

$- 5 x^{2} (x - 1) + x^{4} + 5 x - 6$

Step 3

Factor $x^{4} + 5 x - 6$ using the rational roots test.

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Step 3.1

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.

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

$q = \pm 1$

Step 3.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 3.3

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

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Step 3.3.1

Substitute $1$ into the polynomial.

$1^{4} + 5 \cdot 1 - 6$

Step 3.3.2

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

$1 + 5 \cdot 1 - 6$

Step 3.3.3

Multiply $5$ by $1$ .

$1 + 5 - 6$

Step 3.3.4

Add $1$ and $5$ .

$6 - 6$

Step 3.3.5

Subtract $6$ from $6$ .

$0$

Step 3.4

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.

$\frac{x^{4} + 5 x - 6}{x - 1}$

Step 3.5

Divide $x^{4} + 5 x - 6$ by $x - 1$ .

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Step 3.5.1

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

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

Step 3.5.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

Step 3.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

Step 3.5.4

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

Step 3.5.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

Step 3.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 3.5.7

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

Step 3.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 3.5.9

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 3.5.10

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

Step 3.5.11

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

Step 3.5.12

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

Step 3.5.13

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

$x^{2}$

$x$

Step 3.5.14

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

$x^{2}$

$x$

Step 3.5.15

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

$x^{2}$

$x$

$6 x$

Step 3.5.16

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

$x^{3}$

$x^{2}$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

$x^{2}$

$x$

$6 x$

$6$

Step 3.5.17

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

$x^{3}$

$x^{2}$

$x$

$6$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

$x^{2}$

$x$

$6 x$

$6$

Step 3.5.18

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$x$

$6$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

$x^{2}$

$x$

$6 x$

$6$

$6 x$

$6$

Step 3.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $6 x - 6$

$x^{3}$

$x^{2}$

$x$

$6$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

$x^{2}$

$x$

$6 x$

$6$

$6 x$

$6$

Step 3.5.20

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

$x^{3}$

$x^{2}$

$x$

$6$

$x$

$1$

$x^{4}$

$0 x^{3}$

$0 x^{2}$

$5 x$

$6$

$x^{4}$

$x^{3}$

$0 x^{2}$

$x^{3}$

$x^{2}$

$5 x$

$x^{2}$

$x$

$6 x$

$6$

$6 x$

$6$

$0$

Step 3.5.21

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

$x^{3} + x^{2} + x + 6$

Step 3.6

Write $x^{4} + 5 x - 6$ as a set of factors.

$- 5 x^{2} (x - 1) + (x - 1) (x^{3} + x^{2} + x + 6)$

Step 4

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

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Step 4.1

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

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Step 4.1.1

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.

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

$q = \pm 1$

Step 4.1.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 4.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 4.1.3.1

Substitute $- 2$ into the polynomial.

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

Step 4.1.3.2

Raise $- 2$ to the power of $3$ .

$- 8 + {(- 2)}^{2} - 2 + 6$

Step 4.1.3.3

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

$- 8 + 4 - 2 + 6$

Step 4.1.3.4

Add $- 8$ and $4$ .

$- 4 - 2 + 6$

Step 4.1.3.5

Subtract $2$ from $- 4$ .

$- 6 + 6$

Step 4.1.3.6

Add $- 6$ and $6$ .

$0$

Step 4.1.4

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

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

Step 4.1.5

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

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Step 4.1.5.1

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

$x$

$2$

$x^{3}$

$x^{2}$

$x$

$6$

Step 4.1.5.2

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

					$x^{2}$
	$x$	+	$2$		$x^{3}$	+	$x^{2}$	+	$x$	+	$6$

Step 4.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$2$		$x^{3}$	+	$x^{2}$	+	$x$	+	$6$
			+	$x^{3}$	+	$2 x^{2}$

Step 4.1.5.4

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

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

Step 4.1.5.5

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

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

Step 4.1.5.6

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

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

Step 4.1.5.7

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

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

Step 4.1.5.8

Multiply the new quotient term by the divisor.

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

Step 4.1.5.9

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

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

Step 4.1.5.10

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

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

Step 4.1.5.11

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

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

Step 4.1.5.12

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

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

Step 4.1.5.13

Multiply the new quotient term by the divisor.

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

Step 4.1.5.14

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

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

Step 4.1.5.15

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

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

Step 4.1.5.16

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

$x^{2} - x + 3$

Step 4.1.6

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

$- 5 x^{2} (x - 1) + (x - 1) ((x + 2) (x^{2} - x + 3))$

Step 4.2

Remove unnecessary parentheses.

$- 5 x^{2} (x - 1) + (x - 1) (x + 2) (x^{2} - x + 3)$

Step 5

Factor $x - 1$ out of $- 5 x^{2} (x - 1) + (x - 1) (x + 2) (x^{2} - x + 3)$ .

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Step 5.1

Factor $x - 1$ out of $- 5 x^{2} (x - 1)$ .

$(x - 1) (- 5 x^{2}) + (x - 1) (x + 2) (x^{2} - x + 3)$

Step 5.2

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

$(x - 1) (- 5 x^{2}) + (x - 1) ((x + 2) (x^{2} - x + 3))$

Step 5.3

Factor $x - 1$ out of $(x - 1) (- 5 x^{2}) + (x - 1) ((x + 2) (x^{2} - x + 3))$ .

$(x - 1) (- 5 x^{2} + (x + 2) (x^{2} - x + 3))$

Step 6

Expand $(x + 2) (x^{2} - x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$(x - 1) (- 5 x^{2} + x \cdot x^{2} + x (- x) + x \cdot 3 + 2 x^{2} + 2 (- x) + 2 \cdot 3)$

Step 7

Simplify each term.

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Step 7.1

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

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Step 7.1.1

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

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Step 7.1.1.1

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

$(x - 1) (- 5 x^{2} + x^{1} x^{2} + x (- x) + x \cdot 3 + 2 x^{2} + 2 (- x) + 2 \cdot 3)$

Step 7.1.1.2

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

$(x - 1) (- 5 x^{2} + x^{1 + 2} + x (- x) + x \cdot 3 + 2 x^{2} + 2 (- x) + 2 \cdot 3)$

Step 7.1.2

Add $1$ and $2$ .

$(x - 1) (- 5 x^{2} + x^{3} + x (- x) + x \cdot 3 + 2 x^{2} + 2 (- x) + 2 \cdot 3)$

Step 7.2

Rewrite using the commutative property of multiplication.

$(x - 1) (- 5 x^{2} + x^{3} - x \cdot x + x \cdot 3 + 2 x^{2} + 2 (- x) + 2 \cdot 3)$

Step 7.3

Multiply $x$ by $x$ by adding the exponents.

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Step 7.3.1

Move $x$ .

$(x - 1) (- 5 x^{2} + x^{3} - (x \cdot x) + x \cdot 3 + 2 x^{2} + 2 (- x) + 2 \cdot 3)$

Step 7.3.2

Multiply $x$ by $x$ .

$(x - 1) (- 5 x^{2} + x^{3} - x^{2} + x \cdot 3 + 2 x^{2} + 2 (- x) + 2 \cdot 3)$

Step 7.4

Move $3$ to the left of $x$ .

$(x - 1) (- 5 x^{2} + x^{3} - x^{2} + 3 x + 2 x^{2} + 2 (- x) + 2 \cdot 3)$

Step 7.5

Multiply $- 1$ by $2$ .

$(x - 1) (- 5 x^{2} + x^{3} - x^{2} + 3 x + 2 x^{2} - 2 x + 2 \cdot 3)$

Step 7.6

Multiply $2$ by $3$ .

$(x - 1) (- 5 x^{2} + x^{3} - x^{2} + 3 x + 2 x^{2} - 2 x + 6)$

Step 8

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

$(x - 1) (- 5 x^{2} + x^{3} + x^{2} + 3 x - 2 x + 6)$

Step 9

Subtract $2 x$ from $3 x$ .

$(x - 1) (- 5 x^{2} + x^{3} + x^{2} + x + 6)$

Step 10

Add $- 5 x^{2}$ and $x^{2}$ .

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

Step 11

Factor.

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Step 11.1

Rewrite $x^{3} - 4 x^{2} + x + 6$ in a factored form.

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Step 11.1.1

Factor $x^{3} - 4 x^{2} + x + 6$ using the rational roots test.

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Step 11.1.1.1

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.

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

$q = \pm 1$

Step 11.1.1.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 11.1.1.3

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

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Step 11.1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{3} - 4 {(- 1)}^{2} - 1 + 6$

Step 11.1.1.3.2

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

$- 1 - 4 {(- 1)}^{2} - 1 + 6$

Step 11.1.1.3.3

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

$- 1 - 4 \cdot 1 - 1 + 6$

Step 11.1.1.3.4

Multiply $- 4$ by $1$ .

$- 1 - 4 - 1 + 6$

Step 11.1.1.3.5

Subtract $4$ from $- 1$ .

$- 5 - 1 + 6$

Step 11.1.1.3.6

Subtract $1$ from $- 5$ .

$- 6 + 6$

Step 11.1.1.3.7

Add $- 6$ and $6$ .

$0$

Step 11.1.1.4

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.

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

Step 11.1.1.5

Divide $x^{3} - 4 x^{2} + x + 6$ by $x + 1$ .

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Step 11.1.1.5.1

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

$x$

$1$

$x^{3}$

$4 x^{2}$

$x$

$6$

Step 11.1.1.5.2

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

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

Step 11.1.1.5.3

Multiply the new quotient term by the divisor.

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

Step 11.1.1.5.4

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

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

Step 11.1.1.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$

Step 11.1.1.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$

Step 11.1.1.5.7

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$

Step 11.1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					-	$5 x^{2}$	-	$5 x$

Step 11.1.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} - 5 x$

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$

Step 11.1.1.5.10

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$

Step 11.1.1.5.11

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

				$x^{2}$	-	$5 x$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$

Step 11.1.1.5.12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$5 x$	+	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$

Step 11.1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$5 x$	+	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$
							+	$6 x$	+	$6$

Step 11.1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 6$

				$x^{2}$	-	$5 x$	+	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$
							-	$6 x$	-	$6$

Step 11.1.1.5.15

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

				$x^{2}$	-	$5 x$	+	$6$
$x$	+	$1$		$x^{3}$	-	$4 x^{2}$	+	$x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$5 x^{2}$	+	$x$
					+	$5 x^{2}$	+	$5 x$
							+	$6 x$	+	$6$
							-	$6 x$	-	$6$
										$0$

Step 11.1.1.5.16

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

$x^{2} - 5 x + 6$

Step 11.1.1.6

Write $x^{3} - 4 x^{2} + x + 6$ as a set of factors.

$(x - 1) ((x + 1) (x^{2} - 5 x + 6))$

Step 11.1.2

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 11.1.2.1

Factor $x^{2} - 5 x + 6$ using the AC method.

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Step 11.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 11.1.2.1.2

Write the factored form using these integers.

$(x - 1) ((x + 1) ((x - 3) (x - 2)))$

Step 11.1.2.2

Remove unnecessary parentheses.

$(x - 1) ((x + 1) (x - 3) (x - 2))$

Step 11.2

Remove unnecessary parentheses.

$(x - 1) (x + 1) (x - 3) (x - 2)$