Algebra Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

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

Step 1.2

Factor $2$ out of $6 x - 2$ .

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Step 1.2.1

Factor $2$ out of $6 x$ .

$2 (3 x) - 2 + 3 x^{4} - 4 x^{3} + x^{2} = 0$

Step 1.2.2

Factor $2$ out of $- 2$ .

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

Step 1.2.3

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

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

Step 1.3

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

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Step 1.3.1

Factor $x^{2}$ out of $3 x^{4}$ .

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

Step 1.3.2

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

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

Step 1.3.3

Multiply by $1$ .

$2 (3 x - 1) + x^{2} (3 x^{2}) + x^{2} (- 4 x) + x^{2} \cdot 1 = 0$

Step 1.3.4

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

$2 (3 x - 1) + x^{2} (3 x^{2} - 4 x) + x^{2} \cdot 1 = 0$

Step 1.3.5

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

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

Step 1.4

Factor.

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Step 1.4.1

Factor by grouping.

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Step 1.4.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 1 = 3$ and whose sum is $b = - 4$ .

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Step 1.4.1.1.1

Factor $- 4$ out of $- 4 x$ .

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

Step 1.4.1.1.2

Rewrite $- 4$ as $- 1$ plus $- 3$

$2 (3 x - 1) + x^{2} (3 x^{2} + (- 1 - 3) x + 1) = 0$

Step 1.4.1.1.3

Apply the distributive property.

$2 (3 x - 1) + x^{2} (3 x^{2} - 1 x - 3 x + 1) = 0$

Step 1.4.1.2

Factor out the greatest common factor from each group.

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Step 1.4.1.2.1

Group the first two terms and the last two terms.

$2 (3 x - 1) + x^{2} ((3 x^{2} - 1 x) - 3 x + 1) = 0$

Step 1.4.1.2.2

Factor out the greatest common factor (GCF) from each group.

$2 (3 x - 1) + x^{2} (x (3 x - 1) - (3 x - 1)) = 0$

Step 1.4.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$2 (3 x - 1) + x^{2} ((3 x - 1) (x - 1)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$2 (3 x - 1) + x^{2} (3 x - 1) (x - 1) = 0$

Step 1.5

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

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Step 1.5.1

Factor $3 x - 1$ out of $2 (3 x - 1)$ .

$(3 x - 1) \cdot 2 + x^{2} (3 x - 1) (x - 1) = 0$

Step 1.5.2

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

$(3 x - 1) \cdot 2 + (3 x - 1) (x^{2} (x - 1)) = 0$

Step 1.5.3

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

$(3 x - 1) (2 + x^{2} (x - 1)) = 0$

Step 1.6

Apply the distributive property.

$(3 x - 1) (2 + x^{2} x + x^{2} \cdot - 1) = 0$

Step 1.7

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

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Step 1.7.1

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

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Step 1.7.1.1

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

$(3 x - 1) (2 + x^{2} x + x^{2} \cdot - 1) = 0$

Step 1.7.1.2

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

$(3 x - 1) (2 + x^{2 + 1} + x^{2} \cdot - 1) = 0$

Step 1.7.2

Add $2$ and $1$ .

$(3 x - 1) (2 + x^{3} + x^{2} \cdot - 1) = 0$

Step 1.8

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

$(3 x - 1) (2 + x^{3} - 1 x^{2}) = 0$

Step 1.9

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

$(3 x - 1) (2 + x^{3} - x^{2}) = 0$

Step 1.10

Reorder terms.

$(3 x - 1) (x^{3} - x^{2} + 2) = 0$

Step 1.11

Factor.

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Step 1.11.1

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

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Step 1.11.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 1.11.1.2

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

$\pm 1, \pm 2$

Step 1.11.1.3

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 1.11.1.3.1

Substitute $- 1$ into the polynomial.

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

Step 1.11.1.3.2

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

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

Step 1.11.1.3.3

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

$- 1 - 1 \cdot 1 + 2$

Step 1.11.1.3.4

Multiply $- 1$ by $1$ .

$- 1 - 1 + 2$

Step 1.11.1.3.5

Subtract $1$ from $- 1$ .

$- 2 + 2$

Step 1.11.1.3.6

Add $- 2$ and $2$ .

$0$

Step 1.11.1.4

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.

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

Step 1.11.1.5

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

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Step 1.11.1.5.1

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

$x$

$1$

$x^{3}$

$x^{2}$

$0 x$

$2$

Step 1.11.1.5.2

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

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

Step 1.11.1.5.3

Multiply the new quotient term by the divisor.

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

Step 1.11.1.5.4

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

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

Step 1.11.1.5.5

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

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

Step 1.11.1.5.6

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

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

Step 1.11.1.5.7

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

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

Step 1.11.1.5.8

Multiply the new quotient term by the divisor.

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

Step 1.11.1.5.9

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

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

Step 1.11.1.5.10

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

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

Step 1.11.1.5.11

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

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

Step 1.11.1.5.12

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

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

Step 1.11.1.5.13

Multiply the new quotient term by the divisor.

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

Step 1.11.1.5.14

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

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

Step 1.11.1.5.15

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

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

Step 1.11.1.5.16

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

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

Step 1.11.1.6

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

$(3 x - 1) ((x + 1) (x^{2} - 2 x + 2)) = 0$

Step 1.11.2

Remove unnecessary parentheses.

$(3 x - 1) (x + 1) (x^{2} - 2 x + 2) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 x - 1 = 0$

$x + 1 = 0$

$x^{2} - 2 x + 2 = 0$

Step 3

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 3.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $x^{2} - 2 x + 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} - 2 x + 2$ equal to $0$ .

$x^{2} - 2 x + 2 = 0$

Step 5.2

Solve $x^{2} - 2 x + 2 = 0$ for $x$ .

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Step 5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.2.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 5.2.3.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 5.2.3.1.3

Subtract $8$ from $4$ .

$x = \frac{2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 5.2.3.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 5.2.3.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 5.2.3.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 5.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 5.2.3.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 5.2.3.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 5.2.4

The final answer is the combination of both solutions.

$x = 1 + i, 1 - i$

Step 6

The final solution is all the values that make $(3 x - 1) (x + 1) (x^{2} - 2 x + 2) = 0$ true.

$x = \frac{1}{3}, - 1, 1 + i, 1 - i$

Step 7