div(x-5, (x-1)*(x-2))

1) You can perform a partial fraction decomposition:

\frac{x - 5}{(x - 1) \cdot (x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2}

For the simple factor:

x - 1

and its root:

1

, the unknown A can be easily computed by multiplying by the denominator:

(x - 1) \cdot (x - 2)

and evaluating at the root because the other terms cancel out:

2) Let's solve in

ℝ

(1 - 2) \cdot A = 1 - 5

ⓘ Parentheses are computed first:

1 - 2

-1

■Multiplying by -1 gives the opposite:

-1 \cdot A

- A

■Start from

1

and move back

5

on the number line, going below zero:

1 - 5

-4

- A = -4

ⓘ The sign of both sides is changed

Solution obtained:

A = 4

For the simple factor:

x - 2

and its root:

2

, the unknown B can be easily computed by multiplying by the denominator:

(x - 1) \cdot (x - 2)

and evaluating at the root because the other terms cancel out:

3) Let's solve in

ℝ

(2 - 1) \cdot B = 2 - 5

ⓘ Parentheses are computed first:

2 - 1

1

■Multiplying by 1 changes nothing:

1 \cdot B

B

■Start from

2

and move back

5

on the number line, going below zero:

2 - 5

-3

Solution obtained:

B = -3

4) Finally, you can substitute the values of the unknowns into the initial partial fraction decomposition:

\frac{x - 5}{(x - 1) \cdot (x - 2)} = \frac{4}{x - 1} - \frac{3}{x - 2}

▶Result of the division:

\frac{4}{x - 1} - \frac{3}{x - 2}

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