Algebra Examples

\frac{x}{x^{2} + 3 x + 2} - \frac{1}{(x + 2) (x + 1)}

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

Step 1

Factor

x^{2} + 3 x + 2

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

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Step 1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

2

$2$ and whose sum is

3

$3$ .

1, 2

$1, 2$

Step 1.2

Write the factored form using these integers.

\frac{x}{(x + 1) (x + 2)} - \frac{1}{(x + 2) (x + 1)}

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

\frac{x}{(x + 1) (x + 2)} - \frac{1}{(x + 2) (x + 1)}

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

Step 2

Combine into one fraction.

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Step 2.1

Reorder the factors of

(x + 2) (x + 1)

$(x + 2) (x + 1)$ .

\frac{x}{(x + 1) (x + 2)} - \frac{1}{(x + 1) (x + 2)}

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

Step 2.2

Combine the numerators over the common denominator.

\frac{x - 1}{(x + 1) (x + 2)}

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

\frac{x - 1}{(x + 1) (x + 2)}

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