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31 August, 14:42

The polynomial P (x) = ax³-11x+bx+2, has a factor of x-2 and leaves a reminder of - 12 when divided by x+1. Find the values of a and b

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  1. 31 August, 14:53
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    a = - 5 and b = 30

    Step-by-step explanation:

    Using the Remainder theorem

    If a polynomial p (x) is divided by (x - h) then

    p (h) = Remainder

    If (x - h) is a factor of p (x) then p (h) = 0

    Given (x - 2) is a factor the p (2) = 0

    Given (x + 1) leaves a remainder of - 12 then p ( - 1) = - 12

    Hence

    p (2) = a (2) ³ - 11 (2) + b (2) + 2 = 0, that is

    8a - 22 + 2b + 2 = 0

    8a + 2b - 20 = 0 (add 20 to both sides)

    8a + 2b = 20 → (1)

    and

    p ( - 1) = a ( - 1) ³ - 11 ( - 1) + b ( - 1) + 2 = - 12, that is

    - a + 11 - b + 2 = - 12

    - a - b + 13 = - 12 (subtract 13 from both sides)

    - a - b = - 25 (multiply through by - 1)

    a + b = 25 (subtract b from both sides)

    a = 25 - b → (2)

    Substitute a = 25 - b into (1)

    8 (25 - b) + 2b = 20

    200 - 8b + 2b = 20

    200 - 6b = 20 (subtract 200 from both sides)

    - 6b = - 180 (divide both sides by - 6)

    b = 30

    Substitute b = 30 into (2)

    a = 25 - 30 = - 5
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