Algebra Examples

$\frac{45 p^{5} q^{4} - 60 p^{3} q^{2} - 15 p^{2} q^{3}}{15 p^{2} q^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $15 p^{2} q^{2}$ out of $45 p^{5} q^{4} - 60 p^{3} q^{2} - 15 p^{2} q^{3}$ .

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Step 1.1.1

Factor $15 p^{2} q^{2}$ out of $45 p^{5} q^{4}$ .

$\frac{15 p^{2} q^{2} (3 p^{3} q^{2}) - 60 p^{3} q^{2} - 15 p^{2} q^{3}}{15 p^{2} q^{3}}$

Step 1.1.2

Factor $15 p^{2} q^{2}$ out of $- 60 p^{3} q^{2}$ .

$\frac{15 p^{2} q^{2} (3 p^{3} q^{2}) + 15 p^{2} q^{2} (- 4 p) - 15 p^{2} q^{3}}{15 p^{2} q^{3}}$

Step 1.1.3

Factor $15 p^{2} q^{2}$ out of $- 15 p^{2} q^{3}$ .

$\frac{15 p^{2} q^{2} (3 p^{3} q^{2}) + 15 p^{2} q^{2} (- 4 p) + 15 p^{2} q^{2} (- 1 q)}{15 p^{2} q^{3}}$

Step 1.1.4

Factor $15 p^{2} q^{2}$ out of $15 p^{2} q^{2} (3 p^{3} q^{2}) + 15 p^{2} q^{2} (- 4 p)$ .

$\frac{15 p^{2} q^{2} (3 p^{3} q^{2} - 4 p) + 15 p^{2} q^{2} (- 1 q)}{15 p^{2} q^{3}}$

Step 1.1.5

Factor $15 p^{2} q^{2}$ out of $15 p^{2} q^{2} (3 p^{3} q^{2} - 4 p) + 15 p^{2} q^{2} (- 1 q)$ .

$\frac{15 p^{2} q^{2} (3 p^{3} q^{2} - 4 p - 1 q)}{15 p^{2} q^{3}}$

Step 1.2

Rewrite $- 1 q$ as $- q$ .

$\frac{15 p^{2} q^{2} (3 p^{3} q^{2} - 4 p - q)}{15 p^{2} q^{3}}$

Step 2

Cancel the common factor of $15$ .

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Step 2.1

Cancel the common factor.

$\frac{15 p^{2} q^{2} (3 p^{3} q^{2} - 4 p - q)}{15 p^{2} q^{3}}$

Step 2.2

Rewrite the expression.

$\frac{p^{2} q^{2} (3 p^{3} q^{2} - 4 p - q)}{p^{2} q^{3}}$

Step 3

Cancel the common factor of $p^{2}$ .

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Step 3.1

Cancel the common factor.

$\frac{p^{2} q^{2} (3 p^{3} q^{2} - 4 p - q)}{p^{2} q^{3}}$

Step 3.2

Rewrite the expression.

$\frac{q^{2} (3 p^{3} q^{2} - 4 p - q)}{q^{3}}$

Step 4

Cancel the common factors.

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Step 4.1

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

$\frac{q^{2} (3 p^{3} q^{2} - 4 p - q)}{q^{2} q}$

Step 4.2

Cancel the common factor.

$\frac{q^{2} (3 p^{3} q^{2} - 4 p - q)}{q^{2} q}$

Step 4.3

Rewrite the expression.

$\frac{3 p^{3} q^{2} - 4 p - q}{q}$

Step 5

Split the fraction $\frac{3 p^{3} q^{2} - 4 p - q}{q}$ into two fractions.

$\frac{3 p^{3} q^{2} - 4 p}{q} + \frac{- q}{q}$

Step 6

Split the fraction $\frac{3 p^{3} q^{2} - 4 p}{q}$ into two fractions.

$\frac{3 p^{3} q^{2}}{q} + \frac{- 4 p}{q} + \frac{- q}{q}$

Step 7

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

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Step 7.1

Factor $q$ out of $3 p^{3} q^{2}$ .

$\frac{q (3 p^{3} q)}{q} + \frac{- 4 p}{q} + \frac{- q}{q}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

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

$\frac{q (3 p^{3} q)}{q^{1}} + \frac{- 4 p}{q} + \frac{- q}{q}$

Step 7.2.2

Factor $q$ out of $q^{1}$ .

$\frac{q (3 p^{3} q)}{q \cdot 1} + \frac{- 4 p}{q} + \frac{- q}{q}$

Step 7.2.3

Cancel the common factor.

$\frac{q (3 p^{3} q)}{q \cdot 1} + \frac{- 4 p}{q} + \frac{- q}{q}$

Step 7.2.4

Rewrite the expression.

$\frac{3 p^{3} q}{1} + \frac{- 4 p}{q} + \frac{- q}{q}$

Step 7.2.5

Divide $3 p^{3} q$ by $1$ .

$3 p^{3} q + \frac{- 4 p}{q} + \frac{- q}{q}$

Step 8

Move the negative in front of the fraction.

$3 p^{3} q - \frac{4 p}{q} + \frac{- q}{q}$

Step 9

Cancel the common factor of $q$ .

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Step 9.1

Cancel the common factor.

$3 p^{3} q - \frac{4 p}{q} + \frac{- q}{q}$

Step 9.2

Divide $- 1$ by $1$ .

$3 p^{3} q - \frac{4 p}{q} - 1$