Algebra Examples

$\frac{- 15 w^{2} + 37 w - 8}{- 5 w + 7} + \frac{6 w + 6}{5 w - 7}$

Step 1

Simplify terms.

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Step 1.1

Factor $6$ out of $6 w + 6$ .

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Step 1.1.1

Factor $6$ out of $6 w$ .

$\frac{- 15 w^{2} + 37 w - 8}{- 5 w + 7} + \frac{6 (w) + 6}{5 w - 7}$

Step 1.1.2

Factor $6$ out of $6$ .

$\frac{- 15 w^{2} + 37 w - 8}{- 5 w + 7} + \frac{6 (w) + 6 (1)}{5 w - 7}$

Step 1.1.3

Factor $6$ out of $6 (w) + 6 (1)$ .

$\frac{- 15 w^{2} + 37 w - 8}{- 5 w + 7} + \frac{6 (w + 1)}{5 w - 7}$

Step 1.2

Factor $- 1$ out of $5 w$ .

$\frac{- 15 w^{2} + 37 w - 8}{- 5 w + 7} + \frac{6 (w + 1)}{- (- 5 w) - 7}$

Step 1.3

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{- 15 w^{2} + 37 w - 8}{- 5 w + 7} + \frac{6 (w + 1)}{- (- 5 w) - 1 (7)}$

Step 1.4

Factor $- 1$ out of $- (- 5 w) - 1 (7)$ .

$\frac{- 15 w^{2} + 37 w - 8}{- 5 w + 7} + \frac{6 (w + 1)}{- (- 5 w + 7)}$

Step 1.5

Move a negative from the denominator of $\frac{6 (w + 1)}{- (- 5 w + 7)}$ to the numerator.

$\frac{- 15 w^{2} + 37 w - 8}{- 5 w + 7} + \frac{- (6 (w + 1))}{- 5 w + 7}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{- 15 w^{2} + 37 w - 8 - (6 (w + 1))}{- 5 w + 7}$

Step 2

Simplify the numerator.

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Step 2.1

Multiply $- 1$ by $6$ .

$\frac{- 15 w^{2} + 37 w - 8 - 6 (w + 1)}{- 5 w + 7}$

Step 2.2

Apply the distributive property.

$\frac{- 15 w^{2} + 37 w - 8 - 6 w - 6 \cdot 1}{- 5 w + 7}$

Step 2.3

Multiply $- 6$ by $1$ .

$\frac{- 15 w^{2} + 37 w - 8 - 6 w - 6}{- 5 w + 7}$

Step 2.4

Subtract $6 w$ from $37 w$ .

$\frac{- 15 w^{2} + 31 w - 8 - 6}{- 5 w + 7}$

Step 2.5

Subtract $6$ from $- 8$ .

$\frac{- 15 w^{2} + 31 w - 14}{- 5 w + 7}$

Step 2.6

Factor by grouping.

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Step 2.6.1

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 $a \cdot c = - 15 \cdot - 14 = 210$ and whose sum is $b = 31$ .

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Step 2.6.1.1

Factor $31$ out of $31 w$ .

$\frac{- 15 w^{2} + 31 (w) - 14}{- 5 w + 7}$

Step 2.6.1.2

Rewrite $31$ as $10$ plus $21$

$\frac{- 15 w^{2} + (10 + 21) w - 14}{- 5 w + 7}$

Step 2.6.1.3

Apply the distributive property.

$\frac{- 15 w^{2} + 10 w + 21 w - 14}{- 5 w + 7}$

Step 2.6.2

Factor out the greatest common factor from each group.

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Step 2.6.2.1

Group the first two terms and the last two terms.

$\frac{(- 15 w^{2} + 10 w) + 21 w - 14}{- 5 w + 7}$

Step 2.6.2.2

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

$\frac{5 w (- 3 w + 2) - 7 (- 3 w + 2)}{- 5 w + 7}$

Step 2.6.3

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

$\frac{(- 3 w + 2) (5 w - 7)}{- 5 w + 7}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $5 w - 7$ and $- 5 w + 7$ .

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Step 3.1.1

Factor $- 1$ out of $5 w$ .

$\frac{(- 3 w + 2) (- (- 5 w) - 7)}{- 5 w + 7}$

Step 3.1.2

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{(- 3 w + 2) (- (- 5 w) - 1 (7))}{- 5 w + 7}$

Step 3.1.3

Factor $- 1$ out of $- (- 5 w) - 1 (7)$ .

$\frac{(- 3 w + 2) (- (- 5 w + 7))}{- 5 w + 7}$

Step 3.1.4

Rewrite $- (- 5 w + 7)$ as $- 1 (- 5 w + 7)$ .

$\frac{(- 3 w + 2) (- 1 (- 5 w + 7))}{- 5 w + 7}$

Step 3.1.5

Cancel the common factor.

$\frac{(- 3 w + 2) (- 1 (- 5 w + 7))}{- 5 w + 7}$

Step 3.1.6

Divide $(- 3 w + 2) \cdot (- 1)$ by $1$ .

$(- 3 w + 2) \cdot (- 1)$

Step 3.2

Apply the distributive property.

$- 3 w \cdot - 1 + 2 \cdot - 1$

Step 3.3

Multiply.

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Step 3.3.1

Multiply $- 1$ by $- 3$ .

$3 w + 2 \cdot - 1$

Step 3.3.2

Multiply $2$ by $- 1$ .

$3 w - 2$