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Rajan@2021
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If ax 3 +3x 2 +bx−3 has a factor (2x+3) and leaves remainder −3 when divided by (x+2), find the values of a and b. With these values of a and b, factorise the given expression.

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A basic question from factorisation chapter in which we have been asked to find the values of and b If ax3+3x2+bx3 has a factor (2x+3) and leaves remainder 3 when divided by (x+2),  With these values of a and b, factorise the given expression.

ML Aggarwal(avichal publication), Factorisation, chapter 6, question no 26

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1 Answer

  1. Let
    p(x)=ax3+3x2+bx3
    g(x)=2x+3=0x=23
    f(x)=x+2=0x=2

    Given
    g(x) is a factor of f(x)

      By factor theorem,
    p(23)=0
    a(23)3+3(23)2+b(23)3=0
    a(827)+3(49)+b(23)3=0
    827a+42723b3=0
    8(−27a+5412b)=3
    27a+5412b=24
    3(9a+4b)=2454=30
    9a+4b=10...(i)

    Also, p(x) when divided by f(x) leaves a remainder 3

      By remainder theorem,
    p(2)=3
    a(2)3+3(2)2+b(2)3=3
    8a+122b=0
    8a+2b=12
    4a+b=6...(ii)

    Solving (i)  and  (ii), we get
    a=2  and  b=2

    Hence p(x)=2x3+3x22x3
    =x2(2x+3)(2x+3)==(2x+3)(x21)
    =(2x+3)(x+1)(x1)

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