1EPartialFracsT1


 * Partial Fractions**

Type 1. Linear numerator, quadratic denominator with two factors

math \text {Separate } \dfrac {2x+1}{(x-2)(x-3)} \text { into partial fractions.} math

Divide into two fractions with numerators A and B (constants)

math \text {Let } \dfrac {2x+1}{(x-2)(x-3)} = \dfrac {A}{x-2} + \dfrac {B}{x-3} math

Add the right side together by getting a common denominator math \dfrac {2x+1}{(x-2)(x-3)} = \dfrac {A(x-3) + B(x-2)}{(x-2)(x-3)} math

Equate the numerators math \\ . \qquad 2x + 1 = A(x - 3) + B(x - 2) \\. \\ . \qquad \text {Let } x = 3 \\ . \qquad \Longrightarrow 7 = A(0) + B(1) \\ . \qquad \Longrightarrow B = 7 \\. \\ . \qquad \text {Let } x = 2 \\ . \qquad \Longrightarrow 5 = A(-1) + B(0) \\ . \qquad \Longrightarrow A = -5 math

math \text {Thus } \dfrac {2x+1}{(x-2)(x-3)} = \dfrac {-5}{x-2} + \dfrac {7}{x-3} math

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