Basic Math Examples

$\frac{10 a + 8 - 3 a^{2}}{a^{2} - a - 12} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 1

Factor by grouping.

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Step 1.1

Reorder terms.

$\frac{- 3 a^{2} + 10 a + 8}{a^{2} - a - 12} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 1.2

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 $a \cdot c = - 3 \cdot 8 = - 24$ and whose sum is $b = 10$ .

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Step 1.2.1

Factor $10$ out of $10 a$ .

$\frac{- 3 a^{2} + 10 (a) + 8}{a^{2} - a - 12} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 1.2.2

Rewrite $10$ as $- 2$ plus $12$

$\frac{- 3 a^{2} + (- 2 + 12) a + 8}{a^{2} - a - 12} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 1.2.3

Apply the distributive property.

$\frac{- 3 a^{2} - 2 a + 12 a + 8}{a^{2} - a - 12} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 1.3

Factor out the greatest common factor from each group.

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Step 1.3.1

Group the first two terms and the last two terms.

$\frac{(- 3 a^{2} - 2 a) + 12 a + 8}{a^{2} - a - 12} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 1.3.2

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

$\frac{a (- 3 a - 2) - 4 (- 3 a - 2)}{a^{2} - a - 12} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 1.4

Factor the polynomial by factoring out the greatest common factor, $- 3 a - 2$ .

$\frac{(- 3 a - 2) (a - 4)}{a^{2} - a - 12} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 2

Factor $a^{2} - a - 12$ using the AC method.

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Step 2.1

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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a^{3} - 81 a}{3 a^{2} - 7 a - 6}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $9 a$ out of $9 a^{3} - 81 a$ .

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Step 3.1.1

Factor $9 a$ out of $9 a^{3}$ .

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a^{2}) - 81 a}{3 a^{2} - 7 a - 6}$

Step 3.1.2

Factor $9 a$ out of $- 81 a$ .

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a^{2}) + 9 a (- 9)}{3 a^{2} - 7 a - 6}$

Step 3.1.3

Factor $9 a$ out of $9 a (a^{2}) + 9 a (- 9)$ .

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a^{2} - 9)}{3 a^{2} - 7 a - 6}$

Step 3.2

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

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a^{2} - 3^{2})}{3 a^{2} - 7 a - 6}$

Step 3.3

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

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a + 3) (a - 3)}{3 a^{2} - 7 a - 6}$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 3 \cdot - 6 = - 18$ and whose sum is $b = - 7$ .

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Step 4.1.1

Factor $- 7$ out of $- 7 a$ .

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a + 3) (a - 3)}{3 a^{2} - 7 (a) - 6}$

Step 4.1.2

Rewrite $- 7$ as $2$ plus $- 9$

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a + 3) (a - 3)}{3 a^{2} + (2 - 9) a - 6}$

Step 4.1.3

Apply the distributive property.

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a + 3) (a - 3)}{3 a^{2} + 2 a - 9 a - 6}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a + 3) (a - 3)}{(3 a^{2} + 2 a) - 9 a - 6}$

Step 4.2.2

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

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a + 3) (a - 3)}{a (3 a + 2) - 3 (3 a + 2)}$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $3 a + 2$ .

$\frac{(- 3 a - 2) (a - 4)}{(a - 4) (a + 3)} \cdot \frac{9 a (a + 3) (a - 3)}{(3 a + 2) (a - 3)}$

Step 5

Cancel the common factor of $a + 3$ .

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Step 5.1

Factor $a + 3$ out of $(a - 4) (a + 3)$ .

$\frac{(- 3 a - 2) (a - 4)}{(a + 3) (a - 4)} \cdot \frac{9 a (a + 3) (a - 3)}{(3 a + 2) (a - 3)}$

Step 5.2

Factor $a + 3$ out of $9 a (a + 3) (a - 3)$ .

$\frac{(- 3 a - 2) (a - 4)}{(a + 3) (a - 4)} \cdot \frac{(a + 3) (9 a (a - 3))}{(3 a + 2) (a - 3)}$

Step 5.3

Cancel the common factor.

$\frac{(- 3 a - 2) (a - 4)}{(a + 3) (a - 4)} \cdot \frac{(a + 3) (9 a (a - 3))}{(3 a + 2) (a - 3)}$

Step 5.4

Rewrite the expression.

$\frac{(- 3 a - 2) (a - 4)}{a - 4} \cdot \frac{9 a (a - 3)}{(3 a + 2) (a - 3)}$

Step 6

Multiply $\frac{(- 3 a - 2) (a - 4)}{a - 4}$ by $\frac{9 a (a - 3)}{(3 a + 2) (a - 3)}$ .

$\frac{(- 3 a - 2) (a - 4) (9 a (a - 3))}{(a - 4) ((3 a + 2) (a - 3))}$

Step 7

Cancel the common factor of $a - 4$ .

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Step 7.1

Cancel the common factor.

$\frac{(- 3 a - 2) (a - 4) (9 a (a - 3))}{(a - 4) ((3 a + 2) (a - 3))}$

Step 7.2

Rewrite the expression.

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

Step 8

Cancel the common factor of $a - 3$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Cancel the common factor of $- 3 a - 2$ and $3 a + 2$ .

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Step 9.1

Factor $- 1$ out of $- 3 a$ .

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

Step 9.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{(- (3 a) - 1 (2)) (9 a)}{3 a + 2}$

Step 9.3

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

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

Step 9.4

Rewrite $- (3 a + 2)$ as $- 1 (3 a + 2)$ .

$\frac{- 1 (3 a + 2) (9 a)}{3 a + 2}$

Step 9.5

Cancel the common factor.

$\frac{- 1 (3 a + 2) (9 a)}{3 a + 2}$

Step 9.6

Divide $- 1 (9 a)$ by $1$ .

$- 1 (9 a)$

Step 10

Simplify the expression.

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Step 10.1

Rewrite $- 1 (9 a)$ as $- (9 a)$ .

$- (9 a)$

Step 10.2

Multiply $9$ by $- 1$ .

$- 9 a$