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Rajan@2021
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Use factor theorem to factorize the following polynomials completely. x 3 −13x−12

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In the given polynomial we have been asked to factorise completely
x313x12
ML Aggarwal, Avichal Publication, factorisation, chapter 6, question no 15(ii)

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  1. To solve this type of problem, We have to find out at least one root by hit and trial using the factor theorem.

    Here is a trick:
    If a polynomial function has integer coefficients, then every rational zero will have the form qp, where p is a factor of the constant and q is a factor of the leading coefficient.
    Here
    p=±1,±2,±3,±4,±6,±12 and
    q=±1
    Find every combination of ±qp.
    These are the possible roots of the polynomial function.
    ±1,±2,±3,±4,±6,±12

    Substituting x=1 in f(x), we get
    f(−1)=(−1)313(−1)12=−1+1312=13+13=0
    (x+1) is a factor of f(x)

    Now, dividing f(x) by (x+1), we get
    x313x12=(x+1)(x2x12)
    =(x+1)(x24x+3x12)
    =(x+1){x(x4))+3(x4)}
    =(x+1)(x+3)(x4)

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