1EPartialFracsT3


 * Partial Fractions **

Type 3 Denominator is a Perfect Square

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

The two denominators should be the single term and the squared term math \text {Let } \dfrac {2x-1}{(x+3)^2} = \dfrac {A}{x+3} + \dfrac {B}{(x+3)^2} math

math . \quad \; \dfrac {2x-1}{(x+3)^2} = \dfrac {A(x+3) + B}{(x+3)^2} math

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

Sub B = –7 into the equation and choose ANY other value for x (x = 0 is easy) math . \qquad \text {Let } x = 0 \\ . \qquad \Longrightarrow -1 = A(3) - 7 \\ . \qquad \Longrightarrow A = 2 math

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

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