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Math Help - Factor Theroem

  1. #1
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    Factor Theroem

    Just a quick question about a questoin relating to factor theorem:

    Find the value of k so that when is divided by x + k, the remainder is 3.


    Thanks!
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  2. #2
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    Quote Originally Posted by ty2391 View Post
    Just a quick question about a questoin relating to factor theorem:

    Find the value of k so that when is divided by x + k, the remainder is 3.


    Thanks!
    The division algorithm says we can find the quotient and remainder:

    x^3+5x^2+6x+11 = q(x)(x+k)+3

    Since these are equal, their evaluations are equal as well.
    Let us evaluate this at x=-k.

    Thus,
    (-k)^3+5(-k)^2+6(-k)+11=q(-k)(0)+3
    -k^3+5k-6k+11=3
    k^3-5k+6k-8=0
    Use the rational root theorem and see that k=4 is a solution. (In fact the only one).

    Thus, k=4.
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  3. #3
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    Hello, ty2391!

    Find the value of k so that when x + 5x + 6x + 11 is divided by x + k, the remainder is 3.
    I used the Remainder Theorem:
    . . If f(a) = r, then f(x) (x - a) has a remainder of r.


    We want to find a so that f(a) = 3.

    If a is nonpositive, we can see that f(a) > 11.
    . . Hence, a must be negative.

    We find that: . f(-1) .= .9
    . . . . . . . . . . .f(-2) .= .11
    . . . . . . . . . . .f(-3) .= .13
    . . . . . . . . . . .f(-4) .= . 3 . ← There!

    Therefore: .k = 4

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Check

    (x + 5x + 6x + 11) (x + 4) . = . x + x + 2, .remainder 3

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