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Math Help - [SOLVED] Polynomial factor theorem clarity needed

  1. #1
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    [SOLVED] Polynomial factor theorem clarity needed

    Example
    (x-3) is a factor of f(x)=3x^3-7x^2-5x-3.
    So (x-3) is a factor of 3x^3-7x^2-5x-3=(x-3)(bx^2+cx+d).
    a=3, b=3,c=2,d=1

    I want to clarify my understanding of the below:

    a=3 from 3x^3???
    considering coefficient x^3 gives b=3....from 3x^3???
    considering coefficient x^0 gives -3=-3d..:.d=1. from -3)+d)??? ie the last term??.
    considering coefficient x^2 gives -7=c-3b.:.c=2....why c-3b???
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  2. #2
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    Quote Originally Posted by Neverquit View Post
    Example
    (x-3) is a factor of f(x)=3x^3-7x^2-5x-3.
    So (x-3) is a factor of 3x^3-7x^2-5x-3=(x-3)(bx^2+cx+d).
    a=3, b=3,c=2,d=1
    (x-3) is a factor of 3x^3-7x^2-5x-3
    The other factor is a polynomial (degree = 3-1 = 2)

    Therefore 3x^3-7x^2-5x-3=(x-3)(ax^2+bx+c)
    Expanding gives 3x^3-7x^2-5x-3 = ax^3 + (-3a+b)x^2 + (-3b+c)x - 3c

    Identifying the same power terms
    a=3
    -3a+b=-7
    -3b+c=-5
    -3c=-3

    Therefore
    a=3
    c=1
    b=2
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