1/(x-1)(x-2)+1/(x-2)(x-3)+1/(x-3)(x-4)=2/3

Suzhoumeite

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Jun 15, 2024

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Partial Fraction Expansion

Given the equation:

$\frac{1}{(x-1)(x-2)}+\frac{1}{(x-2)(x-3)}+\frac{1}{(x-3)(x-4)}=\frac{2}{3}$

We can expand the partial fractions to simplify the equation.

Step 1: Partial Fraction Expansion

Let's start by expanding the first fraction:

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

Multiplying both sides by $(x-1)(x-2)$, we get:

$1=A(x-2)+B(x-1)$

Equating coefficients, we get:

$A=-1, B=1$

So,

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

Similarly, let's expand the second fraction:

$\frac{1}{(x-2)(x-3)}=\frac{C}{x-2}+\frac{D}{x-3}$

Multiplying both sides by $(x-2)(x-3)$, we get:

$1=C(x-3)+D(x-2)$

Equating coefficients, we get:

$C=1, D=-1$

So,

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

Finally, let's expand the third fraction:

$\frac{1}{(x-3)(x-4)}=\frac{E}{x-3}+\frac{F}{x-4}$

Multiplying both sides by $(x-3)(x-4)$, we get:

$1=E(x-4)+F(x-3)$

Equating coefficients, we get:

$E=-1, F=1$

So,

$\frac{1}{(x-3)(x-4)}=\frac{-1}{x-3}+\frac{1}{x-4}$

Step 2: Combine the Expanded Fractions

Now, let's combine the expanded fractions:

$\frac{-1}{x-1}+\frac{1}{x-2}+\frac{1}{x-2}-\frac{1}{x-3}-\frac{1}{x-3}+\frac{1}{x-4}=\frac{2}{3}$

Combine like terms:

$\frac{-1}{x-1}+\frac{2}{x-2}-\frac{2}{x-3}+\frac{1}{x-4}=\frac{2}{3}$

Step 3: Simplify the Equation

Simplifying the equation, we get:

$\frac{-1}{x-1}+\frac{2}{x-2}-\frac{2}{x-3}+\frac{1}{x-4}=\frac{2}{3}$

This is the simplified equation.