Basic Math Examples

$\frac{y^{2} - 2 y - 8}{5 y^{3} - 3 y^{2}} \cdot \frac{25 y^{3} - 9 y}{4 y - 16}$

Step 1

Factor $y^{2} - 2 y - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 1.2

Write the factored form using these integers.

$\frac{(y - 4) (y + 2)}{5 y^{3} - 3 y^{2}} \cdot \frac{25 y^{3} - 9 y}{4 y - 16}$

Step 2

Factor $y^{2}$ out of $5 y^{3} - 3 y^{2}$ .

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Step 2.1

Factor $y^{2}$ out of $5 y^{3}$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y) - 3 y^{2}} \cdot \frac{25 y^{3} - 9 y}{4 y - 16}$

Step 2.2

Factor $y^{2}$ out of $- 3 y^{2}$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y) + y^{2} \cdot - 3} \cdot \frac{25 y^{3} - 9 y}{4 y - 16}$

Step 2.3

Factor $y^{2}$ out of $y^{2} (5 y) + y^{2} \cdot - 3$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{25 y^{3} - 9 y}{4 y - 16}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $y$ out of $25 y^{3} - 9 y$ .

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Step 3.1.1

Factor $y$ out of $25 y^{3}$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y (25 y^{2}) - 9 y}{4 y - 16}$

Step 3.1.2

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

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y (25 y^{2}) + y \cdot - 9}{4 y - 16}$

Step 3.1.3

Factor $y$ out of $y (25 y^{2}) + y \cdot - 9$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y (25 y^{2} - 9)}{4 y - 16}$

Step 3.2

Rewrite $25 y^{2}$ as ${(5 y)}^{2}$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y ({(5 y)}^{2} - 9)}{4 y - 16}$

Step 3.3

Rewrite $9$ as $3^{2}$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y ({(5 y)}^{2} - 3^{2})}{4 y - 16}$

Step 3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 y$ and $b = 3$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y (5 y + 3) (5 y - 3)}{4 y - 16}$

Step 4

Simplify terms.

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Step 4.1

Factor $4$ out of $4 y - 16$ .

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Step 4.1.1

Factor $4$ out of $4 y$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y (5 y + 3) (5 y - 3)}{4 (y) - 16}$

Step 4.1.2

Factor $4$ out of $- 16$ .

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y (5 y + 3) (5 y - 3)}{4 y + 4 \cdot - 4}$

Step 4.1.3

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

$\frac{(y - 4) (y + 2)}{y^{2} (5 y - 3)} \cdot \frac{y (5 y + 3) (5 y - 3)}{4 (y - 4)}$

Step 4.2

Combine.

$\frac{(y - 4) (y + 2) (y (5 y + 3) (5 y - 3))}{y^{2} (5 y - 3) (4 (y - 4))}$

Step 4.3

Cancel the common factor of $y - 4$ .

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Step 4.3.1

Cancel the common factor.

$\frac{(y - 4) (y + 2) (y (5 y + 3) (5 y - 3))}{y^{2} (5 y - 3) (4 (y - 4))}$

Step 4.3.2

Rewrite the expression.

$\frac{(y + 2) (y (5 y + 3) (5 y - 3))}{y^{2} (5 y - 3) \cdot (4)}$

Step 4.4

Cancel the common factor of $y$ and $y^{2}$ .

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Step 4.4.1

Factor $y$ out of $(y + 2) (y (5 y + 3) (5 y - 3))$ .

$\frac{y ((y + 2) ((5 y + 3) (5 y - 3)))}{y^{2} (5 y - 3) \cdot (4)}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $y$ out of $y^{2} (5 y - 3) \cdot (4)$ .

$\frac{y ((y + 2) ((5 y + 3) (5 y - 3)))}{y ((y (5 y - 3)) \cdot 4)}$

Step 4.4.2.2

Cancel the common factor.

$\frac{y ((y + 2) ((5 y + 3) (5 y - 3)))}{y ((y (5 y - 3)) \cdot 4)}$

Step 4.4.2.3

Rewrite the expression.

$\frac{(y + 2) ((5 y + 3) (5 y - 3))}{(y (5 y - 3)) \cdot 4}$

Step 4.5

Cancel the common factor of $5 y - 3$ .

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Step 4.5.1

Cancel the common factor.

$\frac{(y + 2) ((5 y + 3) (5 y - 3))}{y (5 y - 3) \cdot 4}$

Step 4.5.2

Rewrite the expression.

$\frac{(y + 2) (5 y + 3)}{(y) \cdot 4}$

Step 4.6

Move $4$ to the left of $y$ .

$\frac{(y + 2) (5 y + 3)}{4 y}$