1EPartialFracsT2


 * Partial Fractions **

Type 2. Numerator is polynomial of degree ≥ denominator

math \text {Express } \dfrac {3x^2 + 3x + 4}{(x-2)(x-3)} \text { into partial fractions.} math

First expand denominator and do long division math \dfrac {3x^2 + 3x + 4}{(x-2)(x-3)} = \dfrac {3x^2 + 3x + 4}{x^2-5x+6} math

math x^2-5x+6 \; \overset{\displaystyle {3} \qquad \qquad .} {\overline { \left) 3x^2+ \;\, 3x+ \; 4 \right. }} math math . \qquad \qquad \quad \; \underline {3x^2-15x+18} math math . \qquad \qquad \qquad \qquad 18x-14 math

math \text {So } \dfrac {3x^2 + 3x + 4}{(x-2)(x-3)} = 3 + \dfrac {18x - 14}{(x-2)(x-3)} math

Now express the fraction part in partial fractions math \text {Let } \dfrac {18x-14}{(x-2)(x-3)} = \dfrac {A}{x-2} + \dfrac {B}{x-3} math

math . \quad \; \dfrac {18x-14}{(x-2)(x-3)} = \dfrac {A(x-3) + B(x-2)}{(x-2)(x-3)} math

Equate the numerators math \\ . \qquad 18x - 14 = A(x - 3) + B(x - 2) \\. \\ . \qquad \text {Let } x = 3 \\ . \qquad \Longrightarrow 40 = A(0) + B(1) \\ . \qquad \Longrightarrow B = 40 \\. \\ . \qquad \text {Let } x = 2 \\ . \qquad \Longrightarrow 22 = A(-1) + B(0) \\ . \qquad \Longrightarrow A = -22 math

math \text {So } \dfrac {18x-14}{(x-2)(x-3)} = \dfrac {-22}{x-2} + \dfrac {40}{x-3} math

math \text {Hence } \dfrac {3x^2 + 3x + 4}{(x-2)(x-3)} = 3 + \dfrac {-22}{x-2} + \dfrac {40}{x-3} math

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