Pre-Algebra Examples

$\frac{\frac{2 y^{2} - 5 y - 3}{2 y^{2} - 3 y - 2}}{\frac{y^{2} - 2 y - 3}{y^{2} - y - 2}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 y^{2} - 5 y - 3}{2 y^{2} - 3 y - 2} \cdot \frac{y^{2} - y - 2}{y^{2} - 2 y - 3}$

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 5$ .

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Step 2.1.1

Factor $- 5$ out of $- 5 y$ .

$\frac{2 y^{2} - 5 (y) - 3}{2 y^{2} - 3 y - 2} \cdot \frac{y^{2} - y - 2}{y^{2} - 2 y - 3}$

Step 2.1.2

Rewrite $- 5$ as $1$ plus $- 6$

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

Step 2.1.3

Apply the distributive property.

$\frac{2 y^{2} + 1 y - 6 y - 3}{2 y^{2} - 3 y - 2} \cdot \frac{y^{2} - y - 2}{y^{2} - 2 y - 3}$

Step 2.1.4

Multiply $y$ by $1$ .

$\frac{2 y^{2} + y - 6 y - 3}{2 y^{2} - 3 y - 2} \cdot \frac{y^{2} - y - 2}{y^{2} - 2 y - 3}$

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.

$\frac{(2 y^{2} + y) - 6 y - 3}{2 y^{2} - 3 y - 2} \cdot \frac{y^{2} - y - 2}{y^{2} - 2 y - 3}$

Step 2.2.2

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

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

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 y + 1$ .

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

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

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Step 3.1.1

Factor $- 3$ out of $- 3 y$ .

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

Step 3.1.2

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

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

Step 3.1.3

Apply the distributive property.

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

Step 3.1.4

Multiply $y$ by $1$ .

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

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

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

Step 3.2.2

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

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

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 y + 1$ .

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

Step 4

Cancel the common factor of $2 y + 1$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\frac{y - 3}{y - 2} \cdot \frac{y^{2} - y - 2}{y^{2} - 2 y - 3}$

Step 5

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

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Step 5.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 5.2

Write the factored form using these integers.

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

Step 6

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

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Step 6.1

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 6.2

Write the factored form using these integers.

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

Step 7

Cancel the common factor of $y - 3$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Cancel the common factor of $y - 2$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Cancel the common factor of $y + 1$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

$1$