Algebra Examples

$\frac{x}{x^{2} + 3 x + 2} + \frac{3}{x + 1}$

Step 1

Factor $x^{2} + 3 x + 2$ using the AC method.

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Step 1.1

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 $3$ .

$1, 2$

Step 1.2

Write the factored form using these integers.

$\frac{x}{(x + 1) (x + 2)} + \frac{3}{x + 1}$

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{3}{x + 1}$ by $\frac{x + 2}{x + 2}$ .

$\frac{x}{(x + 1) (x + 2)} + \frac{3 (x + 2)}{(x + 1) (x + 2)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{x + 3 (x + 2)}{(x + 1) (x + 2)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $3$ by $2$ .

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

Step 4.3

Add $x$ and $3 x$ .

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

Step 4.4

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

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Step 4.4.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) + 6}{(x + 1) (x + 2)}$

Step 4.4.2

Factor $2$ out of $6$ .

$\frac{2 (2 x) + 2 (3)}{(x + 1) (x + 2)}$

Step 4.4.3

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

$\frac{2 (2 x + 3)}{(x + 1) (x + 2)}$