Pre-Algebra Examples

$\frac{y^{2} - 13}{y^{2} - 2 y - 3} - \frac{1}{3 - y}$

Step 1

Factor $y^{2} - 2 y - 3$ using the AC method.

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Step 1.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2

Write the factored form using these integers.

$\frac{y^{2} - 13}{(y - 3) (y + 1)} - \frac{1}{3 - y}$

Step 2

Simplify with factoring out.

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Step 2.1

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

$\frac{y^{2} - 13}{(y - 3) (y + 1)} - \frac{1}{- 1 (- 3) - y}$

Step 2.2

Factor $- 1$ out of $- y$ .

$\frac{y^{2} - 13}{(y - 3) (y + 1)} - \frac{1}{- 1 (- 3) - (y)}$

Step 2.3

Factor $- 1$ out of $- 1 (- 3) - (y)$ .

$\frac{y^{2} - 13}{(y - 3) (y + 1)} - \frac{1}{- 1 (- 3 + y)}$

Step 2.4

Reorder terms.

$\frac{y^{2} - 13}{(y - 3) (y + 1)} - \frac{1}{- 1 (y - 3)}$

Step 3

To write $\frac{y^{2} - 13}{(y - 3) (y + 1)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{y^{2} - 13}{(y - 3) (y + 1)} \cdot \frac{- 1}{- 1} - \frac{1}{- 1 (y - 3)}$

Step 4

To write $- \frac{1}{- 1 (y - 3)}$ as a fraction with a common denominator, multiply by $\frac{y + 1}{y + 1}$ .

$\frac{y^{2} - 13}{(y - 3) (y + 1)} \cdot \frac{- 1}{- 1} - \frac{1}{- 1 (y - 3)} \cdot \frac{y + 1}{y + 1}$

Step 5

Write each expression with a common denominator of $(y - 3) (y + 1) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{y^{2} - 13}{(y - 3) (y + 1)}$ by $\frac{- 1}{- 1}$ .

$\frac{(y^{2} - 13) \cdot - 1}{(y - 3) (y + 1) \cdot - 1} - \frac{1}{- 1 (y - 3)} \cdot \frac{y + 1}{y + 1}$

Step 5.2

Multiply $\frac{1}{- 1 (y - 3)}$ by $\frac{y + 1}{y + 1}$ .

$\frac{(y^{2} - 13) \cdot - 1}{(y - 3) (y + 1) \cdot - 1} - \frac{y + 1}{- 1 (y - 3) (y + 1)}$

Step 5.3

Reorder the factors of $(y - 3) (y + 1) \cdot - 1$ .

$\frac{(y^{2} - 13) \cdot - 1}{- (y + 1) (y - 3)} - \frac{y + 1}{- 1 (y - 3) (y + 1)}$

Step 5.4

Reorder the factors of $- 1 (y - 3) (y + 1)$ .

$\frac{(y^{2} - 13) \cdot - 1}{- (y + 1) (y - 3)} - \frac{y + 1}{- (y + 1) (y - 3)}$

Step 6

Combine the numerators over the common denominator.

$\frac{(y^{2} - 13) \cdot - 1 - (y + 1)}{- (y + 1) (y - 3)}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{y^{2} \cdot - 1 - 13 \cdot - 1 - (y + 1)}{- (y + 1) (y - 3)}$

Step 7.2

Move $- 1$ to the left of $y^{2}$ .

$\frac{- 1 \cdot y^{2} - 13 \cdot - 1 - (y + 1)}{- (y + 1) (y - 3)}$

Step 7.3

Multiply $- 13$ by $- 1$ .

$\frac{- 1 \cdot y^{2} + 13 - (y + 1)}{- (y + 1) (y - 3)}$

Step 7.4

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

$\frac{- y^{2} + 13 - (y + 1)}{- (y + 1) (y - 3)}$

Step 7.5

Apply the distributive property.

$\frac{- y^{2} + 13 - y - 1 \cdot 1}{- (y + 1) (y - 3)}$

Step 7.6

Multiply $- 1$ by $1$ .

$\frac{- y^{2} + 13 - y - 1}{- (y + 1) (y - 3)}$

Step 7.7

Subtract $1$ from $13$ .

$\frac{- y^{2} - y + 12}{- (y + 1) (y - 3)}$

Step 7.8

Factor by grouping.

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Step 7.8.1

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 $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = - 1$ .

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Step 7.8.1.1

Factor $- 1$ out of $- y$ .

$\frac{- y^{2} - (y) + 12}{- (y + 1) (y - 3)}$

Step 7.8.1.2

Rewrite $- 1$ as $3$ plus $- 4$

$\frac{- y^{2} + (3 - 4) y + 12}{- (y + 1) (y - 3)}$

Step 7.8.1.3

Apply the distributive property.

$\frac{- y^{2} + 3 y - 4 y + 12}{- (y + 1) (y - 3)}$

Step 7.8.2

Factor out the greatest common factor from each group.

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Step 7.8.2.1

Group the first two terms and the last two terms.

$\frac{(- y^{2} + 3 y) - 4 y + 12}{- (y + 1) (y - 3)}$

Step 7.8.2.2

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

$\frac{y (- y + 3) + 4 (- y + 3)}{- (y + 1) (y - 3)}$

Step 7.8.3

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

$\frac{(- y + 3) (y + 4)}{- (y + 1) (y - 3)}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Cancel the common factor of $- y + 3$ and $y - 3$ .

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Step 8.1.1

Factor $- 1$ out of $- y$ .

$\frac{(- (y) + 3) (y + 4)}{- (y + 1) (y - 3)}$

Step 8.1.2

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

$\frac{(- (y) - 1 (- 3)) (y + 4)}{- (y + 1) (y - 3)}$

Step 8.1.3

Factor $- 1$ out of $- (y) - 1 (- 3)$ .

$\frac{- (y - 3) (y + 4)}{- (y + 1) (y - 3)}$

Step 8.1.4

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

$\frac{- 1 (y - 3) (y + 4)}{- (y + 1) (y - 3)}$

Step 8.1.5

Cancel the common factor.

$\frac{- 1 (y - 3) (y + 4)}{- (y + 1) (y - 3)}$

Step 8.1.6

Rewrite the expression.

$\frac{- 1 (y + 4)}{- (y + 1)}$

Step 8.2

Dividing two negative values results in a positive value.

$\frac{y + 4}{y + 1}$