Basic Math Examples

$\frac{2 c^{2} + 7 c - 15}{27 - 18 c} \div \frac{c^{4} + c^{3} - 20 c^{2}}{6 c^{3} - 96 c}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 c^{2} + 7 c - 15}{27 - 18 c} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = 7$ .

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Step 2.1.1

Factor $7$ out of $7 c$ .

$\frac{2 c^{2} + 7 (c) - 15}{27 - 18 c} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 2.1.2

Rewrite $7$ as $- 3$ plus $10$

$\frac{2 c^{2} + (- 3 + 10) c - 15}{27 - 18 c} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 2.1.3

Apply the distributive property.

$\frac{2 c^{2} - 3 c + 10 c - 15}{27 - 18 c} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(2 c^{2} - 3 c) + 10 c - 15}{27 - 18 c} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 2.2.2

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

$\frac{c (2 c - 3) + 5 (2 c - 3)}{27 - 18 c} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 2.3

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

$\frac{(2 c - 3) (c + 5)}{27 - 18 c} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3

Simplify terms.

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Step 3.1

Factor $9$ out of $27 - 18 c$ .

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Step 3.1.1

Factor $9$ out of $27$ .

$\frac{(2 c - 3) (c + 5)}{9 (3) - 18 c} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.1.2

Factor $9$ out of $- 18 c$ .

$\frac{(2 c - 3) (c + 5)}{9 (3) + 9 (- 2 c)} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.1.3

Factor $9$ out of $9 (3) + 9 (- 2 c)$ .

$\frac{(2 c - 3) (c + 5)}{9 (3 - 2 c)} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.2

Cancel the common factor of $2 c - 3$ and $3 - 2 c$ .

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Step 3.2.1

Factor $- 1$ out of $2 c$ .

$\frac{(- (- 2 c) - 3) (c + 5)}{9 (3 - 2 c)} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.2.2

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

$\frac{(- (- 2 c) - 1 (3)) (c + 5)}{9 (3 - 2 c)} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.2.3

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

$\frac{- (- 2 c + 3) (c + 5)}{9 (3 - 2 c)} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.2.4

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

$\frac{- 1 (- 2 c + 3) (c + 5)}{9 (3 - 2 c)} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.2.5

Reorder terms.

$\frac{- 1 (- 2 c + 3) (c + 5)}{9 (- 2 c + 3)} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.2.6

Cancel the common factor.

$\frac{- 1 (- 2 c + 3) (c + 5)}{9 (- 2 c + 3)} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.2.7

Rewrite the expression.

$\frac{- 1 (c + 5)}{9} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 3.3

Move the negative in front of the fraction.

$- \frac{c + 5}{9} \cdot \frac{6 c^{3} - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $6 c$ out of $6 c^{3} - 96 c$ .

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Step 4.1.1

Factor $6 c$ out of $6 c^{3}$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c^{2}) - 96 c}{c^{4} + c^{3} - 20 c^{2}}$

Step 4.1.2

Factor $6 c$ out of $- 96 c$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c^{2}) + 6 c (- 16)}{c^{4} + c^{3} - 20 c^{2}}$

Step 4.1.3

Factor $6 c$ out of $6 c (c^{2}) + 6 c (- 16)$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c^{2} - 16)}{c^{4} + c^{3} - 20 c^{2}}$

Step 4.2

Rewrite $16$ as $4^{2}$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c^{2} - 4^{2})}{c^{4} + c^{3} - 20 c^{2}}$

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = c$ and $b = 4$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{4} + c^{3} - 20 c^{2}}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $c^{2}$ out of $c^{4} + c^{3} - 20 c^{2}$ .

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Step 5.1.1

Factor $c^{2}$ out of $c^{4}$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} c^{2} + c^{3} - 20 c^{2}}$

Step 5.1.2

Factor $c^{2}$ out of $c^{3}$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} c^{2} + c^{2} c - 20 c^{2}}$

Step 5.1.3

Factor $c^{2}$ out of $- 20 c^{2}$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} c^{2} + c^{2} c + c^{2} \cdot - 20}$

Step 5.1.4

Factor $c^{2}$ out of $c^{2} c^{2} + c^{2} c$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} (c^{2} + c) + c^{2} \cdot - 20}$

Step 5.1.5

Factor $c^{2}$ out of $c^{2} (c^{2} + c) + c^{2} \cdot - 20$ .

$- \frac{c + 5}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} (c^{2} + c - 20)}$

Step 5.2

Factor $c^{2} + c - 20$ using the AC method.

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Step 5.2.1

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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 5.2.2

Write the factored form using these integers.

$- \frac{c + 5}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} ((c - 4) (c + 5))}$

$- \frac{c + 5}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} (c - 4) (c + 5)}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $c + 5$ .

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Step 6.1.1

Move the leading negative in $- \frac{c + 5}{9}$ into the numerator.

$\frac{- (c + 5)}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} (c - 4) (c + 5)}$

Step 6.1.2

Factor $c + 5$ out of $- (c + 5)$ .

$\frac{(c + 5) \cdot - 1}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} (c - 4) (c + 5)}$

Step 6.1.3

Factor $c + 5$ out of $c^{2} (c - 4) (c + 5)$ .

$\frac{(c + 5) \cdot - 1}{9} \cdot \frac{6 c (c + 4) (c - 4)}{(c + 5) (c^{2} (c - 4))}$

Step 6.1.4

Cancel the common factor.

$\frac{(c + 5) \cdot - 1}{9} \cdot \frac{6 c (c + 4) (c - 4)}{(c + 5) (c^{2} (c - 4))}$

Step 6.1.5

Rewrite the expression.

$\frac{- 1}{9} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} (c - 4)}$

Step 6.2

Cancel the common factor of $3$ .

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Step 6.2.1

Factor $3$ out of $9$ .

$\frac{- 1}{3 (3)} \cdot \frac{6 c (c + 4) (c - 4)}{c^{2} (c - 4)}$

Step 6.2.2

Factor $3$ out of $6 c (c + 4) (c - 4)$ .

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

Step 6.2.3

Cancel the common factor.

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

Step 6.2.4

Rewrite the expression.

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

Step 6.3

Multiply $\frac{- 1}{3}$ by $\frac{2 c (c + 4) (c - 4)}{c^{2} (c - 4)}$ .

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

Step 6.4

Multiply $2$ by $- 1$ .

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

Step 6.5

Cancel the common factor of $c$ and $c^{2}$ .

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Step 6.5.1

Factor $c$ out of $- 2 ((c (c + 4)) (c - 4))$ .

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

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

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

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

Step 6.5.2.2

Cancel the common factor.

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

Step 6.5.2.3

Rewrite the expression.

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

Step 6.6

Cancel the common factor of $c - 4$ .

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Step 6.6.1

Cancel the common factor.

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

Step 6.6.2

Rewrite the expression.

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

Step 6.7

Move the negative in front of the fraction.

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