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8 December, 03:10

find the values os a and b so that the polynomial x3+10x2+ax+b is exactly divisible by (x-1) and (x+2)

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  1. 8 December, 06:40
    0
    a = 7 and b = - 18

    Step-by-step explanation:

    If (x - 1) is a factor of f (x) then x = 1 is a root and f (1) = 0, thus

    f (1) = 1³ + 10 (1) ² + a + b = 0 and

    1 + 10 + a + b = 0 ⇒ a + b = - 11 → (1)

    Similarly if

    (x + 2) is a factor then x = - 2 is a root and f ( - 2) = 0

    f ( - 2) = ( - 2) ³ + 10 ( - 2) ² - 2a + b = 0, thus

    - 8 + 40 - 2a + b = 0 ⇒ - 2a + b = - 32 → (2)

    Solve (1) and (2) simultaneously

    Subtract (1) from (2)

    - 3a = - 21 (divide both sides by 3)

    a = 7

    Substitute a = 7 in (1)

    7 + b = - 11 ⇒ b = - 11 - 7 = - 18

    The required polynomial is

    f (x) = x³ + 10x² + 7x - 18
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