Trigonometry Examples

$\frac{y^{3} - 8}{2 y^{3}} \cdot \frac{4 y}{y^{2} - 5 y + 6}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $8$ as $2^{3}$ .

$\frac{y^{3} - 2^{3}}{2 y^{3}} \cdot \frac{4 y}{y^{2} - 5 y + 6}$

Step 1.2

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

$\frac{(y - 2) (y^{2} + y \cdot 2 + 2^{2})}{2 y^{3}} \cdot \frac{4 y}{y^{2} - 5 y + 6}$

Step 1.3

Simplify.

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Step 1.3.1

Move $2$ to the left of $y$ .

$\frac{(y - 2) (y^{2} + 2 \cdot y + 2^{2})}{2 y^{3}} \cdot \frac{4 y}{y^{2} - 5 y + 6}$

Step 1.3.2

Raise $2$ to the power of $2$ .

$\frac{(y - 2) (y^{2} + 2 y + 4)}{2 y^{3}} \cdot \frac{4 y}{y^{2} - 5 y + 6}$

Step 2

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

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Step 2.1

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 $- 5$ .

$- 3, - 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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Step 3.1

Combine.

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

Step 3.2

Cancel the common factor of $y - 2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $4$ and $2$ .

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Step 3.3.1

Factor $2$ out of $(y^{2} + 2 y + 4) (4 y)$ .

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $y$ and $y^{3}$ .

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Step 3.4.1

Factor $y$ out of $(y^{2} + 2 y + 4) (2 y)$ .

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

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $y$ out of $y^{3} (y - 3)$ .

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

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

Step 3.5

Move $2$ to the left of $y^{2} + 2 y + 4$ .

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