Anonymous
Anonymous asked in Science & MathematicsMathematics · 6 months ago

Express (2x-5)/[(x-2)(3x-1)(2x+5)]?

Given answer is 39/[85(3x-1)] - 1/[45(x-2)] - 40/[153(2x+5)]

Update:

Express (2x-5)/[(x-2)(3x-1)(2x+5)] in Partial Fraction

3 Answers

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  • 6 months ago
    Favorite Answer

    .

    ( 2x - 5 ) / [ ( x - 2 )( 3x - 1 )( 2x + 5 ) ]

    ( 2x - 5 ) = A( 3x - 1 )( 2x + 5 ) + B( x - 2 )( 2x + 5 ) + C( x - 2 )( 3x - 1 )

    ( 2x - 5 ) = A( 6x² + 13x - 5 ) + B( 2x² + x - 10 ) + C( 3x² - 7x + 2 )

    ( 6A + 2B + 3C )x² = 0

    ( 13A + B - 7C )x = 2x

    -5A - 10B + 2C = -5

    A = -1/45, B = 39/85, C = -40/153

  • 6 months ago

    (2x - 5)/[(x-2) (3x-1) (2x+5)]

    = A/(x-2) + B/(3x-1) + C/(2x+5) =>

    2x - 5 = A(6x^2 + 13x - 5) + B(2x^2 + x - 10) + C(3x^2 - 7x + 2) =>

    -5 = -5A - 10B + 2C;

    2 = 13A + B - 7C;

    0 = 6A + 2B + 3C.

    Now you solve the three equations simultaneously the same way you would have done it in high school. I would start by eliminating "B" from the last two equations, obtaining 4 = 20A - 17C. Then by eliminating "B" from the first two equations, obtaining 15 = 125A - 68C. Then eliminate "C" from this pair, obtaining -1 = 45A. I can see my A = -1/45 in the given answer, but I can't guess where the 39 came from, hmm. Anyway, check my work and keep going from there.

  • a/(x - 2) + b/(3x - 1) + c/(2x + 5) = (2x - 5) / ((x - 2) * (3x - 1) * (2x + 5))

    Multiply through by (x - 2) * (3x - 1) * (2x + 5)

    a * (3x - 1) * (2x + 5) + b * (x - 2) * (2x + 5) + c * (x - 2) * (3x - 1) = 0x^2 + 2x - 5

    a * (6x^2 - 2x + 15x - 5) + b * (2x^2 - 4x + 5x - 10) + c * (3x^2 - x - 6x + 2) = 0x^2 + 2x - 5

    a * (6x^2 + 13x - 5) + b * (2x^2 + x - 10) + c * (3x^2 - 7x + 2) = 0x^2 + 2x - 5

    (6a + 2b + 3c) * x^2 + (13a + b - 7c) * x + (-5a - 10b + 2c) = 0x^2 + 2x - 5

    6a + 2b + 3c = 0

    13a + b - 7c = 2

    -5a - 10b + 2c = -5

    6a + 2b + 3c + 2 * (-5a - 10b + 2c) + 13a + b - 7c = 0 + 2 * (-5) + 2

    6a - 10a + 13a + 2b - 20b + b + 3c + 4c - 7c = -10 + 2

    9a - 17b + 0c = -8

    9a - 17b = -8

    9a = 17b - 8

    a = (17b - 8) / 9

    7 * (6a + 2b + 3c) + 3 * (13a + b - 7c) = 7 * 0 + 3 * 2

    42a + 39a + 14b + 3b + 21c - 21c = 0 + 6

    81a + 17b = 6

    81 * (1/9) * (17b - 8) + 17b = 6

    9 * (17b - 8) + 17b = 6

    9 * 17b + 17b - 72 = 6

    (9 + 1) * 17b = 78

    170 * b = 78

    85 * b = 39

    b = 39/85

    a = (17b - 8) / 9

    a = (17 * 39/85 - 8) / 9

    a = (39/5 - 8) / 9

    a = (39/5 - 40/5) / 9

    a = (-1/5) / 9

    a = -1/45

    6a + 2b + 3c = 0

    -6/45 + 78/85 + 3c = 0

    3c = 6/45 - 78/85

    c = 2/45 - 26/85

    c = (1/5) * (2/9 - 26/17)

    c = (2/5) * (1/9 - 13/17)

    c = (2/5) * (17/153 - 117/153)

    c = (2/5) * (-100/153)

    c = -200 / (5 * 153)

    c = -40 / 153

    a = -1/45

    b = 39/85

    c = -40/153

    (-1/45) / (x - 2) + (39/85) / (3x - 1) - (40/153) / (2x + 5) =>

    39 / (85 * (3x - 1)) - 1 / (45 * (x - 2) - 40 / (153 * (2x + 5))

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