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Thread: Factor theorem help

  1. #1
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    Factor theorem help

    I need to factorise this completely, f(x)= x^3+x^2-4x-4 hence I need to solve this equation x^3+x^2-4x-4, can anybody help me factise this please, its just a short question on an engineering course I don't know where to start.
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  2. #2
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    Re: Factor theorem help

    Put simple, the factor theorem states that for a polynomial of degree n

    $\displaystyle x^n+c_{n-1}x^{n-1}+...+c_{2}x^2+c_{1}x+c_{0}$
    (note, the leading coefficient is 1)

    The roots of the polynomial are factors of c0

    I'll demonstrate this for n=3 like your question. With roots a, b and c you should be able to factorise the equation into
    $\displaystyle (x-a)(x-b)(x-c)$

    This expands to

    $\displaystyle x^3+(-a-b-c)x^2+(ab+bc+ac)x-abc$

    You can see that the three roots multiply to give the constant term in the polynomial. So they must be factors of the constant term.

    In your equation -4 is the constant term. The factors are 1,-1,2,-2,4,-4. Try putting each of these factors into the equation and if the result is zero you know its a root. If only one of these is a root you will have to do long division in algebra to get a quadratic which has the other two roots.

    Spoiler:

    It is possible that the roots that multiply to give -4 are $\displaystyle 4^{1/3}, 4^{1/3}, -4^{1/3}$ which are factors of -4 but not whole number factors. When applying the factor theorem we assume that they are whole numbers and hope to get lucky but it isn't always the case. Any question you are assigned to use the factor theorem on will have at least enough whole number roots to reduce the equation to a quadratic
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  3. #3
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    Re: Factor theorem help

    Quote Originally Posted by Andrew187 View Post
    I need to factorise this completely, f(x)= x^3+x^2-4x-4 hence I need to solve this equation x^3+x^2-4x-4, can anybody help me factise this please, its just a short question on an engineering course I don't know where to start.

    It easier to just factor it.
    $\displaystyle x^3+x^2-4x-4=x^2(x+1)-4(x+1)=(x+1)(x+2)(x-2).$
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  4. #4
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    Re: Factor theorem help

    Andrew

    get the free term of the equation -4 and find which numbers divide it . you will find +1,-1,+2,-2,+4,-4 .
    then get each number and substitute in the given polynomial to see which one gives zero .
    if p(x) = x^3+x^2-4x-4 then you will find p(-2)=0 ,p(+2)=0 and p(-1) =0 .
    consequently this polynomial can be factorized : p(x) =(x-2)(x+2)(x+1)
    as simple as such.

    READ CAREFULLY THE INSTRUCTIONS THAT SHAKARRI POSTED BEFORE ME TO HAVE A GENERAL IDEA HOW A POLYNOMIAL OF N DEGREE CAN BE FACTORIZED.

    MINOAS
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