Basic Math Examples

$\frac{9 y - 36}{16 - y^{2}} \cdot \frac{4 y^{2} + 5 y + 1}{36 y + 9}$

Step 1

Factor $9$ out of $9 y - 36$ .

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Step 1.1

Factor $9$ out of $9 y$ .

$\frac{9 (y) - 36}{16 - y^{2}} \cdot \frac{4 y^{2} + 5 y + 1}{36 y + 9}$

Step 1.2

Factor $9$ out of $- 36$ .

$\frac{9 y + 9 \cdot - 4}{16 - y^{2}} \cdot \frac{4 y^{2} + 5 y + 1}{36 y + 9}$

Step 1.3

Factor $9$ out of $9 y + 9 \cdot - 4$ .

$\frac{9 (y - 4)}{16 - y^{2}} \cdot \frac{4 y^{2} + 5 y + 1}{36 y + 9}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{9 (y - 4)}{4^{2} - y^{2}} \cdot \frac{4 y^{2} + 5 y + 1}{36 y + 9}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = y$ .

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{4 y^{2} + 5 y + 1}{36 y + 9}$

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 $a \cdot c = 4 \cdot 1 = 4$ and whose sum is $b = 5$ .

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Step 3.1.1

Factor $5$ out of $5 y$ .

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{4 y^{2} + 5 (y) + 1}{36 y + 9}$

Step 3.1.2

Rewrite $5$ as $1$ plus $4$

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{4 y^{2} + (1 + 4) y + 1}{36 y + 9}$

Step 3.1.3

Apply the distributive property.

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{4 y^{2} + 1 y + 4 y + 1}{36 y + 9}$

Step 3.1.4

Multiply $y$ by $1$ .

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{4 y^{2} + y + 4 y + 1}{36 y + 9}$

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.

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{(4 y^{2} + y) + 4 y + 1}{36 y + 9}$

Step 3.2.2

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

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{y (4 y + 1) + 1 (4 y + 1)}{36 y + 9}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $4 y + 1$ .

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{(4 y + 1) (y + 1)}{36 y + 9}$

Step 4

Factor $9$ out of $36 y + 9$ .

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Step 4.1

Factor $9$ out of $36 y$ .

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{(4 y + 1) (y + 1)}{9 (4 y) + 9}$

Step 4.2

Factor $9$ out of $9$ .

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{(4 y + 1) (y + 1)}{9 (4 y) + 9 (1)}$

Step 4.3

Factor $9$ out of $9 (4 y) + 9 (1)$ .

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{(4 y + 1) (y + 1)}{9 (4 y + 1)}$

Step 5

Cancel the common factor of $9$ .

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Step 5.1

Cancel the common factor.

$\frac{9 (y - 4)}{(4 + y) (4 - y)} \cdot \frac{(4 y + 1) (y + 1)}{9 (4 y + 1)}$

Step 5.2

Rewrite the expression.

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

Step 6

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

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

Step 7

Cancel the common factor of $4 y + 1$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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Step 8.1

Factor $- 1$ out of $y$ .

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

Step 8.2

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 8.3

Factor $- 1$ out of $- 1 (- y) - 1 (4)$ .

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

Step 8.4

Reorder terms.

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

Step 8.5

Cancel the common factor.

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

Step 8.6

Rewrite the expression.

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

Step 9

Move the negative in front of the fraction.

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