Algebra Examples

$(\frac{3}{x} - \frac{x}{3}) (\frac{3 x}{x^{2} + 6 x + 9})$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $9$ as $3^{2}$ .

$(\frac{3}{x} - \frac{x}{3}) \frac{3 x}{x^{2} + 6 x + 3^{2}}$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.3

Rewrite the polynomial.

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

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

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

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

To write $\frac{3}{x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.2

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

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

Step 3.3

Write each expression with a common denominator of $x \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{3}{x}$ by $\frac{3}{3}$ .

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

Step 3.3.2

Multiply $\frac{x}{3}$ by $\frac{x}{x}$ .

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

Step 3.3.3

Reorder the factors of $x \cdot 3$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $3$ by $3$ .

$\frac{\frac{9 - x \cdot x}{3 x} \cdot 3 x}{{(x + 3)}^{2}}$

Step 3.5.2

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

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Step 3.5.2.1

Move $x$ .

$\frac{\frac{9 - (x \cdot x)}{3 x} \cdot 3 x}{{(x + 3)}^{2}}$

Step 3.5.2.2

Multiply $x$ by $x$ .

$\frac{\frac{9 - x^{2}}{3 x} \cdot 3 x}{{(x + 3)}^{2}}$

Step 3.5.3

Rewrite $9 - x^{2}$ in a factored form.

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Step 3.5.3.1

Rewrite $9$ as $3^{2}$ .

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

Step 3.5.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

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

Step 3.6

Combine exponents.

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Step 3.6.1

Combine $\frac{(3 + x) (3 - x)}{3 x}$ and $3$ .

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

Step 3.6.2

Combine $\frac{(3 + x) (3 - x) \cdot 3}{3 x}$ and $x$ .

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

Step 3.7

Reduce the expression $\frac{(3 + x) (3 - x) \cdot 3 x}{3 x}$ by cancelling the common factors.

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Step 3.7.1

Cancel the common factor.

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

Step 3.7.2

Rewrite the expression.

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

Step 3.8

Cancel the common factor of $x$ .

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Step 3.8.1

Cancel the common factor.

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

Step 3.8.2

Divide $(3 + x) (3 - x)$ by $1$ .

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

Step 3.9

Expand $(3 + x) (3 - x)$ using the FOIL Method.

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Step 3.9.1

Apply the distributive property.

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

Step 3.9.2

Apply the distributive property.

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

Step 3.9.3

Apply the distributive property.

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

Step 3.10

Simplify and combine like terms.

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Step 3.10.1

Simplify each term.

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Step 3.10.1.1

Multiply $3$ by $3$ .

$\frac{9 + 3 (- x) + x \cdot 3 + x (- x)}{{(x + 3)}^{2}}$

Step 3.10.1.2

Multiply $- 1$ by $3$ .

$\frac{9 - 3 x + x \cdot 3 + x (- x)}{{(x + 3)}^{2}}$

Step 3.10.1.3

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

$\frac{9 - 3 x + 3 \cdot x + x (- x)}{{(x + 3)}^{2}}$

Step 3.10.1.4

Rewrite using the commutative property of multiplication.

$\frac{9 - 3 x + 3 x - x \cdot x}{{(x + 3)}^{2}}$

Step 3.10.1.5

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

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Step 3.10.1.5.1

Move $x$ .

$\frac{9 - 3 x + 3 x - (x \cdot x)}{{(x + 3)}^{2}}$

Step 3.10.1.5.2

Multiply $x$ by $x$ .

$\frac{9 - 3 x + 3 x - x^{2}}{{(x + 3)}^{2}}$

Step 3.10.2

Add $- 3 x$ and $3 x$ .

$\frac{9 + 0 - x^{2}}{{(x + 3)}^{2}}$

Step 3.10.3

Add $9$ and $0$ .

$\frac{9 - x^{2}}{{(x + 3)}^{2}}$

Step 3.11

Rewrite $9$ as $3^{2}$ .

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

Step 3.12

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

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

Step 4

Cancel the common factor of $3 + x$ and ${(x + 3)}^{2}$ .

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Step 4.1

Reorder terms.

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$\frac{3 - x}{x + 3}$