Algebra Examples

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{x^{2} - 9} + \frac{5 x^{2} + 7 x + 5}{- x^{2} + 9}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify the denominator.

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Step 1.1.1.1

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{x^{2} - 3^{2}} + \frac{5 x^{2} + 7 x + 5}{- x^{2} + 9}$

Step 1.1.1.2

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 $b = 3$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{- x^{2} + 9}$

Step 1.1.2

Simplify the denominator.

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Step 1.1.2.1

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{- x^{2} + 3^{2}}$

Step 1.1.2.2

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{3^{2} - x^{2}}$

Step 1.1.2.3

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 $b = x$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(3 + x) (3 - x)}$

Step 1.2

Simplify terms.

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Step 1.2.1

Reorder terms.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(x + 3) (3 - x)}$

Step 1.2.2

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(x + 3) (- 1 (- 3) - x)}$

Step 1.2.3

Factor $- 1$ out of $- x$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(x + 3) (- 1 (- 3) - (x))}$

Step 1.2.4

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

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{5 x^{2} + 7 x + 5}{(x + 3) (- 1 (- 3 + x))}$

Step 1.2.5

Simplify the expression.

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Step 1.2.5.1

Move a negative from the denominator of $\frac{5 x^{2} + 7 x + 5}{(x + 3) (- 1 (- 3 + x))}$ to the numerator.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{- (5 x^{2} + 7 x + 5)}{(x + 3) (- 3 + x)}$

Step 1.2.5.2

Reorder terms.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58}{(x + 3) (x - 3)} + \frac{- (5 x^{2} + 7 x + 5)}{(x + 3) (x - 3)}$

Step 1.2.6

Combine the numerators over the common denominator.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - (5 x^{2} + 7 x + 5)}{(x + 3) (x - 3)}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the distributive property.

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - (5 x^{2}) - (7 x) - 1 \cdot 5}{(x + 3) (x - 3)}$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $5$ by $- 1$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - 5 x^{2} - (7 x) - 1 \cdot 5}{(x + 3) (x - 3)}$

Step 2.2.2

Multiply $7$ by $- 1$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - 5 x^{2} - 7 x - 1 \cdot 5}{(x + 3) (x - 3)}$

Step 2.2.3

Multiply $- 1$ by $5$ .

$\frac{- 3 x^{3} + 12 x^{2} + 34 x - 58 - 5 x^{2} - 7 x - 5}{(x + 3) (x - 3)}$

Step 2.3

Subtract $5 x^{2}$ from $12 x^{2}$ .

$\frac{- 3 x^{3} + 7 x^{2} + 34 x - 58 - 7 x - 5}{(x + 3) (x - 3)}$

Step 2.4

Subtract $7 x$ from $34 x$ .

$\frac{- 3 x^{3} + 7 x^{2} + 27 x - 58 - 5}{(x + 3) (x - 3)}$

Step 2.5

Subtract $5$ from $- 58$ .

$\frac{- 3 x^{3} + 7 x^{2} + 27 x - 63}{(x + 3) (x - 3)}$

Step 2.6

Rewrite $- 3 x^{3} + 7 x^{2} + 27 x - 63$ in a factored form.

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Step 2.6.1

Factor out the greatest common factor from each group.

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Step 2.6.1.1

Group the first two terms and the last two terms.

$\frac{(- 3 x^{3} + 7 x^{2}) + 27 x - 63}{(x + 3) (x - 3)}$

Step 2.6.1.2

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

$\frac{x^{2} (- 3 x + 7) - 9 (- 3 x + 7)}{(x + 3) (x - 3)}$

Step 2.6.2

Factor the polynomial by factoring out the greatest common factor, $- 3 x + 7$ .

$\frac{(- 3 x + 7) (x^{2} - 9)}{(x + 3) (x - 3)}$

Step 2.6.3

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

$\frac{(- 3 x + 7) (x^{2} - 3^{2})}{(x + 3) (x - 3)}$

Step 2.6.4

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 $b = 3$ .

$\frac{(- 3 x + 7) (x + 3) (x - 3)}{(x + 3) (x - 3)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $x + 3$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(- 3 x + 7) (x + 3) (x - 3)}{(x + 3) (x - 3)}$

Step 3.1.2

Rewrite the expression.

$\frac{(- 3 x + 7) (x - 3)}{x - 3}$

Step 3.2

Cancel the common factor of $x - 3$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(- 3 x + 7) (x - 3)}{x - 3}$

Step 3.2.2

Divide $- 3 x + 7$ by $1$ .

$- 3 x + 7$