Further Partial Fractions

This lesson extends the partial fractions technique to include repeated linear factors and irreducible quadratic factors — cases that go beyond A-Level Mathematics and are required for AQA Further Mathematics.

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Review: Standard Partial Fractions

For distinct linear factors:

$\frac{px + q}{(ax + b)(cx + d)} = \frac{A}{ax + b} + \frac{B}{cx + d}$(ax+b)(cx+d)px+q​=ax+bA​+cx+dB​

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Repeated Linear Factors

If a factor is repeated, we need extra terms:

$\frac{px^2 + qx + r}{(x - a)^2(x - b)} = \frac{A}{x - a} + \frac{B}{(x - a)^2} + \frac{C}{x - b}$(x−a)2(x−b)px2+qx+r​=x−aA​+(x−a)2B​+x−bC​

More generally, a factor $(x - a)^n$(x−a)n contributes $n$n terms:

$\frac{A_1}{x - a} + \frac{A_2}{(x - a)^2} + \cdots + \frac{A_n}{(x - a)^n}$x−aA1​​+(x−a)2A2​​+⋯+(x−a)nAn​​

Worked Example 1: Express $\frac{3x + 5}{(x + 1)^2(x - 2)}$(x+1)2(x−2)3x+5​ in partial fractions.

$\frac{3x + 5}{(x + 1)^2(x - 2)} = \frac{A}{x + 1} + \frac{B}{(x + 1)^2} + \frac{C}{x - 2}$(x+1)2(x−2)3x+5​=x+1A​+(x+1)2B​+x−2C​

Multiply through by $(x + 1)^2(x - 2)$(x+1)2(x−2):

$3x + 5 = A(x + 1)(x - 2) + B(x - 2) + C(x + 1)^2$3x+5=A(x+1)(x−2)+B(x−2)+C(x+1)2

Set $x = -1$x=−1: $2 = B(-3)$2=B(−3), so $B = -2/3$B=−2/3.