Simplify 3/(y^2-3y+2)+5/(y-1)

$\frac{3}{y^{2} - 3 y + 2} + \frac{5}{y - 1}$

Step 1

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.

$\frac{3}{(y - 2) (y - 1)} + \frac{5}{y - 1}$

Step 2

To write $\frac{5}{y - 1}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$\frac{3}{(y - 2) (y - 1)} + \frac{5}{y - 1} \cdot \frac{y - 2}{y - 2}$

Step 3

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

Multiply $\frac{5}{y - 1}$ by $\frac{y - 2}{y - 2}$ .

$\frac{3}{(y - 2) (y - 1)} + \frac{5 (y - 2)}{(y - 1) (y - 2)}$

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

$\frac{3}{(y - 1) (y - 2)} + \frac{5 (y - 2)}{(y - 1) (y - 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 + 5 (y - 2)}{(y - 1) (y - 2)}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{3 + 5 y + 5 \cdot - 2}{(y - 1) (y - 2)}$

Multiply $5$ by $- 2$ .

$\frac{3 + 5 y - 10}{(y - 1) (y - 2)}$

Subtract $10$ from $3$ .

$\frac{5 y - 7}{(y - 1) (y - 2)}$

Simplify 3/(y^2-3y+2)+5/(y-1)

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