#readwise # Partial Fractions - Cover Up Rule  ## Metadata - Author: [[brilliant.org]] - Full Title: Partial Fractions - Cover Up Rule - URL: https://brilliant.org/wiki/partial-fractions-cover-up-rule/ ## Summary The cover-up rule is a method used in partial fraction decomposition to quickly find the coefficients of linear terms. It works when the denominator has linear factors or a mix of linear and irreducible factors. To use this rule, set up the decomposition with one term for each factor in the denominator. This technique helps simplify complex fractions into easier forms for calculation. ## Highlights In partial fraction decomposition, the cover-up rule is a technique to find the coefficients of linear terms in a partial fraction decomposition. It is a faster technique in finding constants in a partial fraction. We can only apply this rule when the denominator is a product of [[Partial Fractions - Linear Factors|linear factors]]. ([View Highlight](https://read.readwise.io/read/01jftdfdgzgw6rtam09ha3cs9y)) --- To compute the coefficients using the cover-up method, first set up a partial fraction decomposition with one term for each of the factors in the denominator. For example, if the denominator has three distinct linear terms, we have the decomposition $ \frac{f(x)}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}. $ ^tb68vv Then by the cover-up method, $A$ can be computed by covering up the term $(x-a)$ in the denominator of the left-hand side and substituting $x=a$ in the remaining expression. This works because the computation is equivalent to multiplying the expression throughout by the term $(x-a)$ and then making the substitution $x=a$. This gives $ A = \frac{f(a)}{(a-b)(a-c)}. $ ^63b4hh Similarly, by substituting $x=b$ and $x=c$, we can compute $B$ and $C$: $B = \frac{f(b)}{(b-a)(b-c)}$$C = \frac{f(c)}{(c-a)(c-b)}$ ^ymbuaa ([View Highlight](https://read.readwise.io/read/01jftdgya89nbcy7nfg750e3gd)) ---