Simplify 3/(y^2-3y+2)+7/(y^2-1)

$\frac{3}{y^{2} - 3 y + 2} + \frac{7}{y^{2} - 1}$

Step 1

Simplify each term.

Factor $y^{2} - 3 y + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$\frac{3}{(y - 2) (y - 1)} + \frac{7}{y^{2} - 1}$

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$\frac{3}{(y - 2) (y - 1)} + \frac{7}{y^{2} - 1^{2}}$

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 $b = 1$ .

$\frac{3}{(y - 2) (y - 1)} + \frac{7}{(y + 1) (y - 1)}$

Step 2

To write $\frac{3}{(y - 2) (y - 1)}$ as a fraction with a common denominator, multiply by $\frac{y + 1}{y + 1}$ .

$\frac{3}{(y - 2) (y - 1)} \cdot \frac{y + 1}{y + 1} + \frac{7}{(y + 1) (y - 1)}$

Step 3

To write $\frac{7}{(y + 1) (y - 1)}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$\frac{3}{(y - 2) (y - 1)} \cdot \frac{y + 1}{y + 1} + \frac{7}{(y + 1) (y - 1)} \cdot \frac{y - 2}{y - 2}$

Step 4

Write each expression with a common denominator of $(y - 2) (y - 1) (y + 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{(y - 2) (y - 1)}$ by $\frac{y + 1}{y + 1}$ .

$\frac{3 (y + 1)}{(y - 2) (y - 1) (y + 1)} + \frac{7}{(y + 1) (y - 1)} \cdot \frac{y - 2}{y - 2}$

Multiply $\frac{7}{(y + 1) (y - 1)}$ by $\frac{y - 2}{y - 2}$ .

$\frac{3 (y + 1)}{(y - 2) (y - 1) (y + 1)} + \frac{7 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

Reorder the factors of $(y - 2) (y - 1) (y + 1)$ .

$\frac{3 (y + 1)}{(y + 1) (y - 1) (y - 2)} + \frac{7 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 (y + 1) + 7 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{3 y + 3 \cdot 1 + 7 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

Multiply $3$ by $1$ .

$\frac{3 y + 3 + 7 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

Apply the distributive property.

$\frac{3 y + 3 + 7 y + 7 \cdot - 2}{(y + 1) (y - 1) (y - 2)}$

Multiply $7$ by $- 2$ .

$\frac{3 y + 3 + 7 y - 14}{(y + 1) (y - 1) (y - 2)}$

Add $3 y$ and $7 y$ .

$\frac{10 y + 3 - 14}{(y + 1) (y - 1) (y - 2)}$

Subtract $14$ from $3$ .

$\frac{10 y - 11}{(y + 1) (y - 1) (y - 2)}$

Simplify 3/(y^2-3y+2)+7/(y^2-1)

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