Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

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 $b = 2$ .

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

Step 2

To write $\frac{3 x}{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Step 3

To write $- \frac{1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{(x + 2) (x - 2)}{(x + 2) (x - 2)}$ .

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

Step 4

Write each expression with a common denominator of $(x + 2) (x - 2) x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3 x}{(x + 2) (x - 2)}$ by $\frac{x^{2}}{x^{2}}$ .

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

Step 4.2

Multiply $\frac{1}{x^{2}}$ by $\frac{(x + 2) (x - 2)}{(x + 2) (x - 2)}$ .

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

Step 4.3

Reorder the factors of $(x + 2) (x - 2) x^{2}$ .

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

Step 4.4

Reorder the factors of $x^{2} ((x + 2) (x - 2))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

Move $x^{2}$ .

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

Step 6.1.2

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

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.3

Add $2$ and $1$ .

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Multiply $- 1$ by $2$ .

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

Step 6.4

Expand $(- x - 2) (x - 2)$ using the FOIL Method.

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Step 6.4.1

Apply the distributive property.

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

Step 6.4.2

Apply the distributive property.

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

Step 6.4.3

Apply the distributive property.

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

Step 6.5

Simplify and combine like terms.

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Step 6.5.1

Simplify each term.

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Step 6.5.1.1

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

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Step 6.5.1.1.1

Move $x$ .

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

Step 6.5.1.1.2

Multiply $x$ by $x$ .

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

Step 6.5.1.2

Multiply $- 2$ by $- 1$ .

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

Step 6.5.1.3

Multiply $- 2$ by $- 2$ .

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

Step 6.5.2

Subtract $2 x$ from $2 x$ .

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

Step 6.5.3

Add $- x^{2}$ and $0$ .

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

Step 6.6

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

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Step 6.6.1

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.

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

$q = \pm 1, \pm 3$

Step 6.6.2

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

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

Step 6.6.3

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 6.6.3.1

Substitute $- 1$ into the polynomial.

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

Step 6.6.3.2

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

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

Step 6.6.3.3

Multiply $3$ by $- 1$ .

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

Step 6.6.3.4

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

$- 3 - 1 \cdot 1 + 4$

Step 6.6.3.5

Multiply $- 1$ by $1$ .

$- 3 - 1 + 4$

Step 6.6.3.6

Subtract $1$ from $- 3$ .

$- 4 + 4$

Step 6.6.3.7

Add $- 4$ and $4$ .

$0$

Step 6.6.4

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.

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

Step 6.6.5

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

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Step 6.6.5.1

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

$x$

$1$

$3 x^{3}$

$x^{2}$

$0 x$

$4$

Step 6.6.5.2

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

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

Step 6.6.5.3

Multiply the new quotient term by the divisor.

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

Step 6.6.5.4

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

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

Step 6.6.5.5

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

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

Step 6.6.5.6

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

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

Step 6.6.5.7

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

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

Step 6.6.5.8

Multiply the new quotient term by the divisor.

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

Step 6.6.5.9

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

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

Step 6.6.5.10

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

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

Step 6.6.5.11

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

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

Step 6.6.5.12

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

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

Step 6.6.5.13

Multiply the new quotient term by the divisor.

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

Step 6.6.5.14

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

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

Step 6.6.5.15

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

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

Step 6.6.5.16

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

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

Step 6.6.6

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

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