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  1. #1
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    theorem

    x^3 + 2x^2 + px -3 is divided by x+1 the remainder is the same as when it is divided by x-2. find value of p


    thanks
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  2. #2
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    Hello,

    Let y be the common remainder. y will be a constant, because you divide by a polynom of degree 1.

    $\displaystyle
    x^3+2x^2+px-3=(x+1)(ax^2+bx+c)+y$
    $\displaystyle
    x^3+2x^2+px-3=(x-2)(a'x^2+b'x+c')+y
    $

    If you develop $\displaystyle (x+1)(ax^2+bx+c)$, you get, by identification :

    $\displaystyle a=1$
    $\displaystyle b=1$
    $\displaystyle c=p-1 (1)$
    $\displaystyle c+y=-3 \Longleftrightarrow y+3=-c$

    (i leave you the right to do the calculus )

    In the same way,
    $\displaystyle a'=1$
    $\displaystyle b'=4$
    $\displaystyle c'=p+8 (2)$
    $\displaystyle -2c'+y=-3 \Longleftrightarrow y+3=2c'$

    Hence 2c'=-c.

    With (1) and (2), we can state that 2(p+8)=-(p-1)

    And then, just solve for p
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  3. #3
    Super Member wingless's Avatar
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    Let P(x) and Q(x) be polynomials and k be the root of Q(x)=0.

    Then the remainder of the division P(x)/Q(x) is equal to P(k).


    In your question,
    $\displaystyle P(x) = x^3 + 2x^2 + px - 3$ and $\displaystyle Q(x)=x+1$
    The root of Q(x) = 0 is,
    $\displaystyle x+1=0$, $\displaystyle x=-1$

    So find P(-1),
    $\displaystyle P(x) = x^3 + 2x^2 + px - 3$
    $\displaystyle P(-1) = -1 + 2 -p - 3 = -p-2$


    Now find the remainder for $\displaystyle Q(x) = x-2$
    The root of $\displaystyle Q(x) = 0$ is,
    $\displaystyle x-2 =0$, $\displaystyle x=2$

    So $\displaystyle P(2) = 8 + 8 + 2p -3 = 2p + 13$

    Since the remainders are the same,
    $\displaystyle -p-2 = 2p + 13$
    $\displaystyle p=-5$
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  4. #4
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    Quote Originally Posted by gracey View Post
    x^3 + 2x^2 + px -3 is divided by x+1 the remainder is the same as when it is divided by x-2. find value of p


    thanks
    To streamline things a bit ......

    Use the remainder theorem:

    Let the remainder be r. Then:

    -1 + 2 - p - 3 = r => -2 - p = r .... (1)

    8 + 8 + 2p - 3 = r => 13 + 2p = r .... (2)

    Equate (1) and (2): -2 - p = 13 + 2p => p = -5.
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