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Thread: Difficulty Solving 2 Questions (Remainder Theorem Lesson)

  1. #1
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    Difficulty Solving 2 Questions (Remainder Theorem Lesson)

    Hi everyone!
    I'm having a bit of difficulty solving these two questions.
    1.) When the polynomial mx3 - 3x2 + n + 2 is divided by x + 3, the remainder is -1. When it is divided by x - 2, the remainder is -4. Determine the values of m and n.
    For this one, I made the equations -27m - 27 - 3n + 2 = -1, and 8m - 12 + 2n + 2 = -4. Then, I multiplied the second equation by -3/2 to remove n via elimination.
    -27m - 3n = 24
    -12m - 3n = -9
    -15m = 33
    m = -11/5
    The texbook says this is the correct answer, and I am unable to figure out what to do from here to get the value of n. For reference, the textbook says it is 59/5.

    2.) When 3x2 + 10x - 3 is divided by x + k, the remainder is 5. Determine the values of k.
    I'm just completely lost with this one. This is the first question I've seen with the polynomial being divided by 2 unknown variables.

    Any help is appreciated, thank you!

    (Also, apologies for formatting sloppiness, I'm new here )
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  2. #2
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    Re: Difficulty Solving 2 Questions (Remainder Theorem Lesson)

    Quote Originally Posted by hungryjimmy View Post
    Hi everyone!
    I'm having a bit of difficulty solving these two questions.
    1.) When the polynomial mx3 - 3x2 + n + 2 is divided by x + 3, the remainder is -1. When it is divided by x - 2, the remainder is -4. Determine the values of m and n.
    For this one, I made the equations -27m - 27 - 3n + 2 = -1, and 8m - 12 + 2n + 2 = -4. Then, I multiplied the second equation by -3/2 to remove n via elimination.
    -27m - 3n = 24
    -12m - 3n = -9
    -15m = 33
    m = -11/5
    The texbook says this is the correct answer, and I am unable to figure out what to do from here to get the value of n. For reference, the textbook says it is 59/5.
    For this first problem, you're missing an "x" in the polynomial. You treated it as if it were there when you solved for the correct m-value according to the book.

    The polynomial should be $\displaystyle \ \ mx^3 - 3x^2 + nx + 2 \ \ $ instead. If you were to go back and substitute m = -11/5 (the textbook's answer) and solve for n
    in the equation 8m - 12 + 2n + 2 = -4 (given above), you will see that the value of n will equal 59/5, as the textbook mentions. (I worked this out.)

    ________________________________________


    Attempt at edit:

    For the second problem, do as you did for the first one:

    Because you are dividing by the factor (x + k), then you would correspondingly substitute in (-k):

    $\displaystyle 3(-k)^2 + 10(-k) - 3 = 5$

    $\displaystyle 3k^2 - 10k - 3 = 5$

    $\displaystyle 3k^2 - 10k - 8 = 0 $


    Now you can solve this quadratic equation for the two values of k. (It happens to factor with integer coefficients.)
    Last edited by greg1313; Sep 18th 2018 at 01:04 PM.
    Thanks from MarkFL and hungryjimmy
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    Re: Difficulty Solving 2 Questions (Remainder Theorem Lesson)

    Quote Originally Posted by hungryjimmy View Post
    -12m - 3n = -9

    m = -11/5
    Subs. -11/5 in equation and you'll get n = 59/5
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