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Math Help - [SOLVED] polynomial quesstion

  1. #1
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    [SOLVED] polynomial quesstion

    hello all,
    i hope this question is in the right board, anyhow, here is my problem:

    if  x^3+px+q is divisible by both x-3 and x+2, find the values of p and q


    thanks
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  2. #2
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    don't worry guys, after some more work i got the answer myself, here is the working out for anybody interested:

    P(x) = x^3+px+q
    if both x-3 and x+2 are factors of P(x), then P(3) = 0 = P(-2)

    therefore
    (3)^3+p3+q = (-2)^3+p2+q
    simplifying  27+3p = -8-2p
         -35 = 5p
            p = -7

    therefore P(x)=(x-3)(x+2)(x+k)

    now, (k*-1) + (k*-6) =7
    k=1

    therefore P(x)=(x-3)(x+2)(x+1)
    expanding P(x)=x^3-7x-6

    therefore q=-6
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  3. #3
    Super Member dhiab's Avatar
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    Quote Originally Posted by str33tl0rd View Post
    hello all,
    i hope this question is in the right board, anyhow, here is my problem:

    if  x^3+px+q is divisible by both x-3 and x+2, find the values of p and q


    thanks
    Hello :There is the solution
    I'have :

    \begin{array}{l}<br />
x^2 + px + q = \left( {x - 3} \right)\left( {x + 2} \right)\left( {ax + b} \right) \\ <br />
x^2 + px + q = ax^3 + \left( {b - a} \right)x^2 + \left( { - b - 6a} \right)x - 6b \\ <br />
\end{array}
    I'HAVE THE SYSTEM :
    \left\{ \begin{array}{l}<br />
a = 1 \\ <br />
b - a = 0 \\ <br />
- b - 6a = p \\ <br />
- 6b = q \\ <br />
\end{array} \right.<br />


    a = 1,b =1
    Subst :
    CONCLUSION
    \left\{ \begin{array}{l}<br />
q = - 6 \\ <br />
p = - 7 \\ <br />
\end{array} \right.<br />
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  4. #4
    Senior Member pacman's Avatar
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    "if is divisible by both and , find the values of p and q."

    Not so ELEGANT SOLUTION:

    Notice that x^3 + px + q = (x - a)(x - b)(x - c) = (x - 3)(x + 2)(x - k)

    that the coefficient of x^2 is ZERO, that means the sum of its roots is ZERO.

    then, -(-3 + 2 - k) = 0, or

    3 - 2 + k = 0,

    k = -1; thus the 3 roots are, -2, -1 and 3.

    p = (-2)(-1) + (-2)(3) + (-1)(3) = 2 - 6 - 3 = 2 - 9 = -7

    q = -(-2)(-1)(3) = -6

    x^3 -7x - 6 = (x - 3)(x + 2)(x + 1).





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