Simplify (3y)/(y^2-17y+72)+12/(y^2-81)-11/(y^2+y-72)

$\frac{3 y}{y^{2} - 17 y + 72} + \frac{12}{y^{2} - 81} - \frac{11}{y^{2} + y - 72}$

Simplify each term.

Factor $y^{2} - 17 y + 72$ using the AC method.

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 $72$ and whose sum is $- 17$ .

$- 9, - 8$

Write the factored form using these integers.

$\frac{3 y}{(y - 9) (y - 8)} + \frac{12}{y^{2} - 81} - \frac{11}{y^{2} + y - 72}$

Simplify the denominator.

Rewrite $81$ as $9^{2}$ .

$\frac{3 y}{(y - 9) (y - 8)} + \frac{12}{y^{2} - 9^{2}} - \frac{11}{y^{2} + y - 72}$

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 $b = 9$ .

$\frac{3 y}{(y - 9) (y - 8)} + \frac{12}{(y + 9) (y - 9)} - \frac{11}{y^{2} + y - 72}$

Factor $y^{2} + y - 72$ using the AC method.

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 $- 72$ and whose sum is $1$ .

$- 8, 9$

Write the factored form using these integers.

$\frac{3 y}{(y - 9) (y - 8)} + \frac{12}{(y + 9) (y - 9)} - \frac{11}{(y - 8) (y + 9)}$

Find the common denominator.

Multiply $\frac{3 y}{(y - 9) (y - 8)}$ by $\frac{y + 9}{y + 9}$ .

$\frac{3 y}{(y - 9) (y - 8)} \cdot \frac{y + 9}{y + 9} + \frac{12}{(y + 9) (y - 9)} - \frac{11}{(y - 8) (y + 9)}$

Multiply $\frac{3 y}{(y - 9) (y - 8)}$ by $\frac{y + 9}{y + 9}$ .

$\frac{3 y (y + 9)}{(y - 9) (y - 8) (y + 9)} + \frac{12}{(y + 9) (y - 9)} - \frac{11}{(y - 8) (y + 9)}$

Multiply $\frac{12}{(y + 9) (y - 9)}$ by $\frac{y - 8}{y - 8}$ .

$\frac{3 y (y + 9)}{(y - 9) (y - 8) (y + 9)} + \frac{12}{(y + 9) (y - 9)} \cdot \frac{y - 8}{y - 8} - \frac{11}{(y - 8) (y + 9)}$

Multiply $\frac{12}{(y + 9) (y - 9)}$ by $\frac{y - 8}{y - 8}$ .

$\frac{3 y (y + 9)}{(y - 9) (y - 8) (y + 9)} + \frac{12 (y - 8)}{(y + 9) (y - 9) (y - 8)} - \frac{11}{(y - 8) (y + 9)}$

Multiply $\frac{11}{(y - 8) (y + 9)}$ by $\frac{y - 9}{y - 9}$ .

$\frac{3 y (y + 9)}{(y - 9) (y - 8) (y + 9)} + \frac{12 (y - 8)}{(y + 9) (y - 9) (y - 8)} - (\frac{11}{(y - 8) (y + 9)} \cdot \frac{y - 9}{y - 9})$

Multiply $\frac{11}{(y - 8) (y + 9)}$ by $\frac{y - 9}{y - 9}$ .

$\frac{3 y (y + 9)}{(y - 9) (y - 8) (y + 9)} + \frac{12 (y - 8)}{(y + 9) (y - 9) (y - 8)} - \frac{11 (y - 9)}{(y - 8) (y + 9) (y - 9)}$

Reorder the factors of $(y + 9) (y - 9) (y - 8)$ .

$\frac{3 y (y + 9)}{(y - 9) (y - 8) (y + 9)} + \frac{12 (y - 8)}{(y + 9) (y - 8) (y - 9)} - \frac{11 (y - 9)}{(y - 8) (y + 9) (y - 9)}$

Reorder the factors of $(y - 8) (y + 9) (y - 9)$ .

$\frac{3 y (y + 9)}{(y - 9) (y - 8) (y + 9)} + \frac{12 (y - 8)}{(y + 9) (y - 8) (y - 9)} - \frac{11 (y - 9)}{(y + 9) (y - 8) (y - 9)}$

Combine the numerators over the common denominator.

$\frac{3 y (y + 9) + 12 (y - 8) - 11 (y - 9)}{(y + 9) (y - 8) (y - 9)}$

Simplify each term.

Apply the distributive property.

$\frac{3 y \cdot y + 3 y \cdot 9 + 12 (y - 8) - 11 (y - 9)}{(y + 9) (y - 8) (y - 9)}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$\frac{3 (y \cdot y) + 3 y \cdot 9 + 12 (y - 8) - 11 (y - 9)}{(y + 9) (y - 8) (y - 9)}$

Multiply $y$ by $y$ .

$\frac{3 y^{2} + 3 y \cdot 9 + 12 (y - 8) - 11 (y - 9)}{(y + 9) (y - 8) (y - 9)}$

Multiply $9$ by $3$ .

$\frac{3 y^{2} + 27 y + 12 (y - 8) - 11 (y - 9)}{(y + 9) (y - 8) (y - 9)}$

Apply the distributive property.

$\frac{3 y^{2} + 27 y + 12 y + 12 \cdot - 8 - 11 (y - 9)}{(y + 9) (y - 8) (y - 9)}$

Multiply $12$ by $- 8$ .

$\frac{3 y^{2} + 27 y + 12 y - 96 - 11 (y - 9)}{(y + 9) (y - 8) (y - 9)}$

Apply the distributive property.

$\frac{3 y^{2} + 27 y + 12 y - 96 - 11 y - 11 \cdot - 9}{(y + 9) (y - 8) (y - 9)}$

Multiply $- 11$ by $- 9$ .

$\frac{3 y^{2} + 27 y + 12 y - 96 - 11 y + 99}{(y + 9) (y - 8) (y - 9)}$

Simplify by adding terms.

Add $27 y$ and $12 y$ .

$\frac{3 y^{2} + 39 y - 96 - 11 y + 99}{(y + 9) (y - 8) (y - 9)}$

Subtract $11 y$ from $39 y$ .

$\frac{3 y^{2} + 28 y - 96 + 99}{(y + 9) (y - 8) (y - 9)}$

Add $- 96$ and $99$ .

$\frac{3 y^{2} + 28 y + 3}{(y + 9) (y - 8) (y - 9)}$

Simplify (3y)/(y^2-17y+72)+12/(y^2-81)-11/(y^2+y-72)

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