Precalculus Examples

$\frac{- 8 r^{3} s - 12 r^{2} s^{2} + 20 r s^{3}}{- 4 r s}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4 r s$ out of $- 8 r^{3} s - 12 r^{2} s^{2} + 20 r s^{3}$ .

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Step 1.1.1

Factor $4 r s$ out of $- 8 r^{3} s$ .

$\frac{4 r s (- 2 r^{2}) - 12 r^{2} s^{2} + 20 r s^{3}}{- 4 r s}$

Step 1.1.2

Factor $4 r s$ out of $- 12 r^{2} s^{2}$ .

$\frac{4 r s (- 2 r^{2}) + 4 r s (- 3 r s) + 20 r s^{3}}{- 4 r s}$

Step 1.1.3

Factor $4 r s$ out of $20 r s^{3}$ .

$\frac{4 r s (- 2 r^{2}) + 4 r s (- 3 r s) + 4 r s (5 s^{2})}{- 4 r s}$

Step 1.1.4

Factor $4 r s$ out of $4 r s (- 2 r^{2}) + 4 r s (- 3 r s)$ .

$\frac{4 r s (- 2 r^{2} - 3 r s) + 4 r s (5 s^{2})}{- 4 r s}$

Step 1.1.5

Factor $4 r s$ out of $4 r s (- 2 r^{2} - 3 r s) + 4 r s (5 s^{2})$ .

$\frac{4 r s (- 2 r^{2} - 3 r s + 5 s^{2})}{- 4 r s}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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 $a \cdot c = - 2 \cdot 5 = - 10$ and whose sum is $b = - 3$ .

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Step 1.2.1.1

Reorder terms.

$\frac{4 r s (- 2 r^{2} + 5 s^{2} - 3 r s)}{- 4 r s}$

Step 1.2.1.2

Reorder $5 s^{2}$ and $- 3 r s$ .

$\frac{4 r s (- 2 r^{2} - 3 r s + 5 s^{2})}{- 4 r s}$

Step 1.2.1.3

Factor $- 3$ out of $- 3 r s$ .

$\frac{4 r s (- 2 r^{2} - 3 (r s) + 5 s^{2})}{- 4 r s}$

Step 1.2.1.4

Rewrite $- 3$ as $2$ plus $- 5$

$\frac{4 r s (- 2 r^{2} + (2 - 5) (r s) + 5 s^{2})}{- 4 r s}$

Step 1.2.1.5

Apply the distributive property.

$\frac{4 r s (- 2 r^{2} + 2 (r s) - 5 (r s) + 5 s^{2})}{- 4 r s}$

Step 1.2.1.6

Move parentheses.

$\frac{4 r s (- 2 r^{2} + 2 r s - 5 r s + 5 s^{2})}{- 4 r s}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{4 r s ((- 2 r^{2} + 2 r s) - 5 r s + 5 s^{2})}{- 4 r s}$

Step 1.2.2.2

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

$\frac{4 r s (2 r (- r + s) + 5 s (- r + s))}{- 4 r s}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $- r + s$ .

$\frac{4 r s ((- r + s) (2 r + 5 s))}{- 4 r s}$

$\frac{4 r s (- r + s) (2 r + 5 s)}{- 4 r s}$

Step 2

Cancel the common factor of $4$ and $- 4$ .

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Step 2.1

Factor $4$ out of $4 r s (- r + s) (2 r + 5 s)$ .

$\frac{4 ((r s (- r + s)) (2 r + 5 s))}{- 4 r s}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $4$ out of $- 4 r s$ .

$\frac{4 ((r s (- r + s)) (2 r + 5 s))}{4 (- r s)}$

Step 2.2.2

Cancel the common factor.

$\frac{4 ((r s (- r + s)) (2 r + 5 s))}{4 (- r s)}$

Step 2.2.3

Rewrite the expression.

$\frac{(r s (- r + s)) (2 r + 5 s)}{- r s}$

Step 3

Cancel the common factor of $r$ .

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Step 3.1

Cancel the common factor.

$\frac{r s (- r + s) (2 r + 5 s)}{- r s}$

Step 3.2

Rewrite the expression.

$\frac{(s (- r + s)) (2 r + 5 s)}{- 1 s}$

Step 4

Cancel the common factor of $s$ .

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Step 4.1

Cancel the common factor.

$\frac{s (- r + s) (2 r + 5 s)}{- 1 s}$

Step 4.2

Rewrite the expression.

$\frac{(- r + s) (2 r + 5 s)}{- 1}$

Step 4.3

Move the negative one from the denominator of $\frac{(- r + s) (2 r + 5 s)}{- 1}$ .

$- 1 \cdot ((- r + s) (2 r + 5 s))$

Step 5

Rewrite $- 1 \cdot ((- r + s) (2 r + 5 s))$ as $- ((- r + s) (2 r + 5 s))$ .

$- ((- r + s) (2 r + 5 s))$

Step 6

Expand $(- r + s) (2 r + 5 s)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$- (- r (2 r + 5 s) + s (2 r + 5 s))$

Step 6.2

Apply the distributive property.

$- (- r (2 r) - r (5 s) + s (2 r + 5 s))$

Step 6.3

Apply the distributive property.

$- (- r (2 r) - r (5 s) + s (2 r) + s (5 s))$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite using the commutative property of multiplication.

$- (- 1 \cdot 2 r \cdot r - r (5 s) + s (2 r) + s (5 s))$

Step 7.1.2

Multiply $r$ by $r$ by adding the exponents.

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Step 7.1.2.1

Move $r$ .

$- (- 1 \cdot 2 (r \cdot r) - r (5 s) + s (2 r) + s (5 s))$

Step 7.1.2.2

Multiply $r$ by $r$ .

$- (- 1 \cdot 2 r^{2} - r (5 s) + s (2 r) + s (5 s))$

Step 7.1.3

Multiply $- 1$ by $2$ .

$- (- 2 r^{2} - r (5 s) + s (2 r) + s (5 s))$

Step 7.1.4

Rewrite using the commutative property of multiplication.

$- (- 2 r^{2} - 1 \cdot 5 r s + s (2 r) + s (5 s))$

Step 7.1.5

Multiply $- 1$ by $5$ .

$- (- 2 r^{2} - 5 r s + s (2 r) + s (5 s))$

Step 7.1.6

Rewrite using the commutative property of multiplication.

$- (- 2 r^{2} - 5 r s + 2 s r + s (5 s))$

Step 7.1.7

Rewrite using the commutative property of multiplication.

$- (- 2 r^{2} - 5 r s + 2 s r + 5 s \cdot s)$

Step 7.1.8

Multiply $s$ by $s$ by adding the exponents.

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Step 7.1.8.1

Move $s$ .

$- (- 2 r^{2} - 5 r s + 2 s r + 5 (s \cdot s))$

Step 7.1.8.2

Multiply $s$ by $s$ .

$- (- 2 r^{2} - 5 r s + 2 s r + 5 s^{2})$

Step 7.2

Add $- 5 r s$ and $2 s r$ .

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Step 7.2.1

Move $s$ .

$- (- 2 r^{2} - 5 r s + 2 r s + 5 s^{2})$

Step 7.2.2

Add $- 5 r s$ and $2 r s$ .

$- (- 2 r^{2} - 3 r s + 5 s^{2})$

Step 8

Apply the distributive property.

$- (- 2 r^{2}) - (- 3 r s) - (5 s^{2})$

Step 9

Simplify.

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Step 9.1

Multiply $- 2$ by $- 1$ .

$2 r^{2} - (- 3 r s) - (5 s^{2})$

Step 9.2

Multiply $- 3$ by $- 1$ .

$2 r^{2} + 3 (r s) - (5 s^{2})$

Step 9.3

Multiply $5$ by $- 1$ .

$2 r^{2} + 3 (r s) - 5 s^{2}$

$2 r^{2} + 3 r s - 5 s^{2}$