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}

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

Step 1

Factor

y^{2} - 2 y - 8

$y^{2} - 2 y - 8$ using the AC method.

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Step 1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 8

$- 8$ and whose sum is

- 2

$- 2$ .

- 4, 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}

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

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

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

Step 2

Factor

y^{2}

$y^{2}$ out of

5 y^{3} - 3 y^{2}

$5 y^{3} - 3 y^{2}$ .

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Step 2.1

Factor

y^{2}

$y^{2}$ out of

5 y^{3}

$5 y^{3}$ .

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

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

Step 2.2

Factor

y^{2}

$y^{2}$ out of

- 3 y^{2}

$- 3 y^{2}$ .

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

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

Step 2.3

Factor

y^{2}

$y^{2}$ out of

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

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

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

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

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

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

Step 3

Simplify the numerator.

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Step 3.1

Factor

y

$y$ out of

25 y^{3} - 9 y

$25 y^{3} - 9 y$ .

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Step 3.1.1

Factor

y

$y$ out of

25 y^{3}

$25 y^{3}$ .

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

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

Step 3.1.2

Factor

y

$y$ out of

- 9 y

$- 9 y$ .

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

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

Step 3.1.3

Factor

y

$y$ out of

y (25 y^{2}) + y \cdot - 9

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

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

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

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

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

Step 3.2

Rewrite

25 y^{2}

$25 y^{2}$ as

{(5 y)}^{2}

${(5 y)}^{2}$ .

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

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

Step 3.3

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

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

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

Step 3.4

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 5 y

$a = 5 y$ and

b = 3

$b = 3$ .

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

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

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

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

Step 4

Simplify terms.

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Step 4.1

Factor

4

$4$ out of

4 y - 16

$4 y - 16$ .

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Step 4.1.1

Factor

4

$4$ out of

4 y

$4 y$ .

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

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

Step 4.1.2

Factor

4

$4$ out of

- 16

$- 16$ .

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

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

Step 4.1.3

Factor

4

$4$ out of

4 y + 4 \cdot - 4

$4 y + 4 \cdot - 4$ .

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

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

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

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

Step 4.3

Cancel the common factor of

y - 4

$y - 4$ .

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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

$y$ .

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