Simplify 4/(y^2-3y+2)+6/(y^2-1)

$\frac{4}{y^{2} - 3 y + 2} + \frac{6}{y^{2} - 1}$

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.

$\frac{4}{(y - 2) (y - 1)} + \frac{6}{y^{2} - 1}$

Simplify the denominator.

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

$\frac{4}{(y - 2) (y - 1)} + \frac{6}{y^{2} - 1^{2}}$

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 $b = 1$ .

$\frac{4}{(y - 2) (y - 1)} + \frac{6}{(y + 1) (y - 1)}$

Step 2

To write $\frac{4}{(y - 2) (y - 1)}$ as a fraction with a common denominator, multiply by $\frac{y + 1}{y + 1}$ .

$\frac{4}{(y - 2) (y - 1)} \cdot \frac{y + 1}{y + 1} + \frac{6}{(y + 1) (y - 1)}$

Step 3

To write $\frac{6}{(y + 1) (y - 1)}$ as a fraction with a common denominator, multiply by $\frac{y - 2}{y - 2}$ .

$\frac{4}{(y - 2) (y - 1)} \cdot \frac{y + 1}{y + 1} + \frac{6}{(y + 1) (y - 1)} \cdot \frac{y - 2}{y - 2}$

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 $\frac{4}{(y - 2) (y - 1)}$ by $\frac{y + 1}{y + 1}$ .

$\frac{4 (y + 1)}{(y - 2) (y - 1) (y + 1)} + \frac{6}{(y + 1) (y - 1)} \cdot \frac{y - 2}{y - 2}$

Multiply $\frac{6}{(y + 1) (y - 1)}$ by $\frac{y - 2}{y - 2}$ .

$\frac{4 (y + 1)}{(y - 2) (y - 1) (y + 1)} + \frac{6 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

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

$\frac{4 (y + 1)}{(y + 1) (y - 1) (y - 2)} + \frac{6 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 (y + 1) + 6 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

Step 6

Simplify the numerator.

Factor $2$ out of $4 (y + 1) + 6 (y - 2)$ .

Factor $2$ out of $4 (y + 1)$ .

$\frac{2 (2 (y + 1)) + 6 (y - 2)}{(y + 1) (y - 1) (y - 2)}$

Factor $2$ out of $6 (y - 2)$ .

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

Factor $2$ out of $2 (2 (y + 1)) + 2 (3 (y - 2))$ .

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

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

Apply the distributive property.

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

Multiply $3$ by $- 2$ .

$\frac{2 (2 y + 2 + 3 y - 6)}{(y + 1) (y - 1) (y - 2)}$

Add $2 y$ and $3 y$ .

$\frac{2 (5 y + 2 - 6)}{(y + 1) (y - 1) (y - 2)}$

Subtract $6$ from $2$ .

$\frac{2 (5 y - 4)}{(y + 1) (y - 1) (y - 2)}$

Simplify 4/(y^2-3y+2)+6/(y^2-1)

Related Questions

Support LCMGCF.com by disabling your adblocker.