Cover-Up Rule

The Cover-Up Rule

The cover-up rule (also called the Heaviside cover-up method) is a quick way to find coefficients when decomposing rational functions with linear factors only.

The Method

For \(\frac{P(x)}{(x-a)(x-b)(x-c)}\):

1. Write the decomposition as \(\frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}\)
2. To find \(A\): cover up \((x-a)\) in the denominator, substitute \(x=a\) into what's left
3. Repeat for \(B\) and \(C\)

Example 1: Simple Linear Factors

Find the partial fraction decomposition of \(\frac{5x+3}{(x-1)(x+3)}\):

To find \(A\): cover up \((x-1)\), substitute \(x=1\):

\(A = \frac{5(1)+3}{1+3} = \frac{8}{4} = 2\)

To find \(B\): cover up \((x+3)\), substitute \(x=-3\):

\(B = \frac{5(-3)+3}{-3-1} = \frac{-15+3}{-4} = \frac{-12}{-4} = 3\)

Example 2: Three Linear Factors

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

\(A = \frac{2(2)^2 + 2 - 1}{(2+1)(2-1)} = \frac{8+2-1}{3 \cdot 1} = \frac{9}{3} = 3\) \(B = \frac{2(-1)^2 + (-1) - 1}{(-1-2)(-1-1)} = \frac{2 -1 -1}{(-3)(-2)} = \frac{0}{6} = 0\) \(C = \frac{2(1)^2 + 1 - 1}{(1-2)(1+1)} = \frac{2+1-1}{(-1)(2)} = \frac{2}{-2} = -1\)

Limitations

The cover-up rule only works for:

• Linear factors in the denominator
• Numerators that are constants (not polynomials with \(x\))

For repeated factors or quadratic factors, use the standard method of multiplying and equating coefficients.