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Show that 2x+7 is a factor of 2x³+5x²-11x-14, hence factorise completely using factor theorem

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An important question from factorisation chapter in which we are to Show that 2x+7 is a factor of 2x³+5x²-11x-14 and eventually we need to factorize completely.

ML Aggarwal, Avichal Publication, chapter 6, Factorisation, Question no 14

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  1. If  2x + 7 in factor  of 2x3 + 5x2 – 11x – 14

    then on putting  2x + 7 = 0

    x = – 7/2

    f(-7/2) = 0

    = 2(- 7/2)3 + 5(- 7/2)2 – 11(- 7/2) – 14

    = – 343/4 + 245/4 + 77/2 – 14

    = – 399/4 + 245 + 154/4

    = – 399 + 399/4 = 0

    Hence  2x + 7 is one factor.

    Now  2x3 + 5x2 – 11x – 14

    = x2(2x + 7) – x(2x + 7) – 2(2x + 7)

    = (2x + 7)(x2 – x – 2)

    = (2x + 7)(x2 + x – 2x – 2)

    = (2x + 7)[x(x + 1) – 2(x + 1)]

    = (2x + 7)(x – 2)(x + 1)

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