Finite Math Examples

$\frac{x^{2} + 2 x y + y^{2}}{x^{2} - 2 x y + y^{2}} \cdot \frac{9 x - 9 y}{5 x + 5 y}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$2 x y = 2 \cdot x \cdot y$

Step 1.2

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot y + y^{2}}{x^{2} - 2 x y + y^{2}} \cdot \frac{9 x - 9 y}{5 x + 5 y}$

Step 1.3

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

$\frac{{(x + y)}^{2}}{x^{2} - 2 x y + y^{2}} \cdot \frac{9 x - 9 y}{5 x + 5 y}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$2 x y = 2 \cdot x \cdot y$

Step 2.2

Rewrite the polynomial.

$\frac{{(x + y)}^{2}}{x^{2} - 2 \cdot x \cdot y + y^{2}} \cdot \frac{9 x - 9 y}{5 x + 5 y}$

Step 2.3

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

$\frac{{(x + y)}^{2}}{{(x - y)}^{2}} \cdot \frac{9 x - 9 y}{5 x + 5 y}$

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

Factor $9$ out of $9 x$ .

$\frac{{(x + y)}^{2}}{{(x - y)}^{2}} \cdot \frac{9 (x) - 9 y}{5 x + 5 y}$

Step 3.1.2

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

$\frac{{(x + y)}^{2}}{{(x - y)}^{2}} \cdot \frac{9 (x) + 9 (- y)}{5 x + 5 y}$

Step 3.1.3

Factor $9$ out of $9 (x) + 9 (- y)$ .

$\frac{{(x + y)}^{2}}{{(x - y)}^{2}} \cdot \frac{9 (x - y)}{5 x + 5 y}$

Step 3.2

Factor $5$ out of $5 x + 5 y$ .

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Step 3.2.1

Factor $5$ out of $5 x$ .

$\frac{{(x + y)}^{2}}{{(x - y)}^{2}} \cdot \frac{9 (x - y)}{5 (x) + 5 y}$

Step 3.2.2

Factor $5$ out of $5 y$ .

$\frac{{(x + y)}^{2}}{{(x - y)}^{2}} \cdot \frac{9 (x - y)}{5 (x) + 5 (y)}$

Step 3.2.3

Factor $5$ out of $5 (x) + 5 (y)$ .

$\frac{{(x + y)}^{2}}{{(x - y)}^{2}} \cdot \frac{9 (x - y)}{5 (x + y)}$

Step 3.3

Combine.

$\frac{{(x + y)}^{2} (9 (x - y))}{{(x - y)}^{2} (5 (x + y))}$

Step 3.4

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

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Step 3.4.1

Factor $x + y$ out of ${(x + y)}^{2} (9 (x - y))$ .

$\frac{(x + y) ((x + y) (9 (x - y)))}{{(x - y)}^{2} (5 (x + y))}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $x + y$ out of ${(x - y)}^{2} (5 (x + y))$ .

$\frac{(x + y) ((x + y) (9 (x - y)))}{(x + y) ({(x - y)}^{2} \cdot (5))}$

Step 3.4.2.2

Cancel the common factor.

$\frac{(x + y) ((x + y) (9 (x - y)))}{(x + y) ({(x - y)}^{2} \cdot (5))}$

Step 3.4.2.3

Rewrite the expression.

$\frac{(x + y) (9 (x - y))}{{(x - y)}^{2} \cdot (5)}$

Step 3.5

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

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Step 3.5.1

Factor $x - y$ out of $(x + y) (9 (x - y))$ .

$\frac{(x - y) ((x + y) \cdot (9))}{{(x - y)}^{2} \cdot (5)}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

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

$\frac{(x - y) ((x + y) \cdot (9))}{(x - y) ((x - y) \cdot 5)}$

Step 3.5.2.2

Cancel the common factor.

$\frac{(x - y) ((x + y) \cdot (9))}{(x - y) ((x - y) \cdot 5)}$

Step 3.5.2.3

Rewrite the expression.

$\frac{(x + y) \cdot (9)}{(x - y) \cdot 5}$

Step 3.6

Reorder.

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Step 3.6.1

Move $9$ to the left of $x + y$ .

$\frac{9 \cdot (x + y)}{(x - y) \cdot 5}$

Step 3.6.2

Move $5$ to the left of $x - y$ .

$\frac{9 (x + y)}{5 (x - y)}$