Simplify 3/(y+5)+y/(y^2+7y+10)

$\frac{3}{y + 5} + \frac{y}{y^{2} + 7 y + 10}$

Step 1

Factor $y^{2} + 7 y + 10$ 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 $10$ and whose sum is $7$ .

$2, 5$

Write the factored form using these integers.

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

Step 2

To write $\frac{3}{y + 5}$ as a fraction with a common denominator, multiply by $\frac{y + 2}{y + 2}$ .

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

Step 3

Write each expression with a common denominator of $(y + 5) (y + 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{y + 5}$ by $\frac{y + 2}{y + 2}$ .

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

Reorder the factors of $(y + 5) (y + 2)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $3$ by $2$ .

$\frac{3 y + 6 + y}{(y + 2) (y + 5)}$

Add $3 y$ and $y$ .

$\frac{4 y + 6}{(y + 2) (y + 5)}$

Factor $2$ out of $4 y + 6$ .

Factor $2$ out of $4 y$ .

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

Factor $2$ out of $6$ .

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

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

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

Simplify 3/(y+5)+y/(y^2+7y+10)

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