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Math Help - Need help on factor theorem.

  1. #1
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    Unhappy Need help on factor theorem.

    I am supposed to use the Factor Theorem to either factor the polynomial completely or to prove that it has no linear factors with integer coefficients.
    I think I haven't completely understood the whole concept yet, which is why I couldn't get my brain working on the following problems.

    a. P(x)= x^4+4x^3-7x^2-34x-24
    b. P(x)= x^6-6x^5+15x^4-20x^3+15x^2-6x+1

    I think these are harder than the regular ones like P(x)=x^3+2x^2-9x+3, etc.

    Thanks.
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  2. #2
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    Hello zxazsw
    Quote Originally Posted by zxazsw View Post
    I am supposed to use the Factor Theorem to either factor the polynomial completely or to prove that it has no linear factors with integer coefficients.
    I think I haven't completely understood the whole concept yet, which is why I couldn't get my brain working on the following problems.

    a. P(x)= x^4+4x^3-7x^2-34x-24
    b. P(x)= x^6-6x^5+15x^4-20x^3+15x^2-6x+1

    I think these are harder than the regular ones like P(x)=x^3+2x^2-9x+3, etc.

    Thanks.
    Two things you need to use here:

    • The Factor Theorem itself: (x-a) is a factor of P(x) if and only if P(a) = 0.


    • If the constant term in P(x) is k, then in any factor of the form (ax+b), b k. must be a factor of

    So in part (a), you'll only need to try values of x which are factors of 24 (not forgetting to try negative numbers as well), to see whether P(x) = 0. If my arithmetic is correct, none of them works, so there aren't any linear factors with integer coefficients.

    By the same reasoning, in part (b), (x \pm1) are the only possibilities. So try x =  1 first. And, even when you have found the first factor, keep on trying with the quotient until you're sure there are no factors left. (Hint: do you know anything about Pascal's triangle or the Binomial Theorem?)

    Grandad
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