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    Remainder Theorem to Solve a Polynomial

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    1) Determine the value of k so that when P(x) = x(cubed) + kx(squared) - 2x(squared) + 1. Is divided by x + 2, the remainder is 5

    2) Using long division, determine the remainder when P(x) is divided by x-3

    © BrainMass Inc. brainmass.com December 24, 2021, 11:42 pm ad1c9bdddf
    https://brainmass.com/math/computing-values-of-functions/remainder-theorem-solve-polynomial-593851

    SOLUTION This solution is FREE courtesy of BrainMass!

    1) Determine the value of k so that when P(x) = x(cubed) + kx(squared) - 2x(squared) + 1 is divided by x + 2, the remainder is 5.

    2) Using long division, determine the remainder when P(x) is divided by x-3

    The polynomial will not be, in general, in the form you gave. I am taking the general form

    P(x) = x^4 + kx^2 - 2x^2 + 1 leaves remainder 5 when divided by x+2

    Refer to Remainder theorem which states that the remainder when x-a divides a polynomial P(x) is equal to P(a)

    So, we should have P(-2) = 5 -> (-2)^4 + k(-2)^2 - 2(-2)^2 + 1 = 5
    That gives 16 +4k - 8 + 1 = 5
    This gives k = -1

    Now, the polynomial is P(x) = x^4 - x^2 - 2x^2 + 1 = x^4 -3x^2 + 1
    Using the long division,

    x - 3)x^4 - 3x^2 + 1(x^3 + 3x^2 + 6x + 17
    x^4 - 3x^3
    _______________
    3x^3 - 3x^2
    3x^3 - 9x^2
    ________________
    6x^2 + 1
    6x^2 - 18x
    ___________
    18x + 1
    18x - 54
    ________
    55

    Hence the remainder is 55
    This can be verified P(3) = (3)^4 - 3(3)^2 + 1 = 81 - 27 + 1 = 55

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:42 pm ad1c9bdddf>
    https://brainmass.com/math/computing-values-of-functions/remainder-theorem-solve-polynomial-593851

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