Basic Math Examples

$\frac{y^{2} - y - 6}{2 y^{2} + 13 y + 6} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

Step 1

Factor $y^{2} - y - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.2

Write the factored form using these integers.

$\frac{(y - 3) (y + 2)}{2 y^{2} + 13 y + 6} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

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 $a \cdot c = 2 \cdot 6 = 12$ and whose sum is $b = 13$ .

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Step 2.1.1

Factor $13$ out of $13 y$ .

$\frac{(y - 3) (y + 2)}{2 y^{2} + 13 (y) + 6} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

Step 2.1.2

Rewrite $13$ as $1$ plus $12$

$\frac{(y - 3) (y + 2)}{2 y^{2} + (1 + 12) y + 6} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

Step 2.1.3

Apply the distributive property.

$\frac{(y - 3) (y + 2)}{2 y^{2} + 1 y + 12 y + 6} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

Step 2.1.4

Multiply $y$ by $1$ .

$\frac{(y - 3) (y + 2)}{2 y^{2} + y + 12 y + 6} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

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.

$\frac{(y - 3) (y + 2)}{(2 y^{2} + y) + 12 y + 6} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

Step 2.2.2

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

$\frac{(y - 3) (y + 2)}{y (2 y + 1) + 6 (2 y + 1)} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

Step 2.3

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

$\frac{(y - 3) (y + 2)}{(2 y + 1) (y + 6)} \cdot \frac{2 y^{2} - 11 y - 6}{y^{2} + 5 y + 6}$

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 11$ .

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Step 3.1.1

Factor $- 11$ out of $- 11 y$ .

$\frac{(y - 3) (y + 2)}{(2 y + 1) (y + 6)} \cdot \frac{2 y^{2} - 11 (y) - 6}{y^{2} + 5 y + 6}$

Step 3.1.2

Rewrite $- 11$ as $1$ plus $- 12$

$\frac{(y - 3) (y + 2)}{(2 y + 1) (y + 6)} \cdot \frac{2 y^{2} + (1 - 12) y - 6}{y^{2} + 5 y + 6}$

Step 3.1.3

Apply the distributive property.

$\frac{(y - 3) (y + 2)}{(2 y + 1) (y + 6)} \cdot \frac{2 y^{2} + 1 y - 12 y - 6}{y^{2} + 5 y + 6}$

Step 3.1.4

Multiply $y$ by $1$ .

$\frac{(y - 3) (y + 2)}{(2 y + 1) (y + 6)} \cdot \frac{2 y^{2} + y - 12 y - 6}{y^{2} + 5 y + 6}$

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.

$\frac{(y - 3) (y + 2)}{(2 y + 1) (y + 6)} \cdot \frac{(2 y^{2} + y) - 12 y - 6}{y^{2} + 5 y + 6}$

Step 3.2.2

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

$\frac{(y - 3) (y + 2)}{(2 y + 1) (y + 6)} \cdot \frac{y (2 y + 1) - 6 (2 y + 1)}{y^{2} + 5 y + 6}$

Step 3.3

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

$\frac{(y - 3) (y + 2)}{(2 y + 1) (y + 6)} \cdot \frac{(2 y + 1) (y - 6)}{y^{2} + 5 y + 6}$

Step 4

Factor $y^{2} + 5 y + 6$ using the AC method.

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Step 4.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $y + 2$ .

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Step 5.1.1

Factor $y + 2$ out of $(y - 3) (y + 2)$ .

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

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

Step 5.2

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

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

Multiply $\frac{y - 3}{y + 6}$ by $\frac{y - 6}{y + 3}$ .

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