Algebra Examples

$\frac{y}{3 y^{2} - 12} \cdot (y^{2} - 4 y + 4)$

Step 1

Simplify the denominator.

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Step 1.1

Factor $3$ out of $3 y^{2} - 12$ .

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Step 1.1.1

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

$\frac{y}{3 (y^{2}) - 12} \cdot (y^{2} - 4 y + 4)$

Step 1.1.2

Factor $3$ out of $- 12$ .

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

Step 1.1.3

Factor $3$ out of $3 y^{2} + 3 \cdot - 4$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Factor using the perfect square rule.

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Step 3.1

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

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

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 y = 2 \cdot y \cdot 2$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 2$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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