How Do You Decompose a Rational Expression Into Partial Fractions?

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Multiply every term by the common denominator to obtain the identity

This step is valid for all permissible values of $x$, so the numerators form a polynomial identity. Choosing a root is therefore not a numerical approximation; it is an efficient way to isolate one unknown coefficient exactly.

Find $A$ and $B$ by substituting the roots $x=a$ and $x=b$. Each substitution makes one term zero. Alternatively, expand the right-hand side and compare coefficients of corresponding powers of $x$.

Worked Example

Decompose

Start with

Clearing the denominator gives

Set $x=1$:

Set $x=-2$:

Therefore,

Verification confirms the result:

Watch the signs: at $x=-2$, $x-1=-3$. Substitute the constants back into the identity to catch errors before writing the final decomposition.

A common misconception is to place the original numerator over each factor. This is incorrect: because each denominator factor is linear, its corresponding numerator must be an unknown constant.

Exam Technique

This HL-only topic may appear on Paper 1 or Paper 2, often as preparation for integration in AHL 5.15. Show the proposed decomposition, the cleared-denominator identity, and the calculation of every constant. A correct final expression without supporting working may not receive full marks. Recombine the fractions as a quick algebraic check. On Paper 1, keep the algebra exact; on Paper 2, technology may check but not replace the setup.