Mathematics

Question

It is given a polynomial f(x) = 2x³ + ax² - bx + 12, where a and b are constants. When f(x) is divided by x-2, the remainder is -18. It is known that f(x) is divisible by x + 4. a) Find the values of a and b b) Solve the equation f(x) = 0​

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  • Answer:

    Step-by-step explanation:

    (a). 2( 2 )³ + a ( 2 )² - b ( 2 ) + 12 + ( 18 ) = 0

    16 + 4a - 2b + 30 = 0

    4a - 2b = - 46

    2a - b = - 23 ....... ( 1 )

    2 ( - 4 )³ + a ( - 4 )² - b ( - 4 ) + 12 = 0

    - 128 + 16a + 4b + 12 = 0

    16a + 4b = 116

    4a + b = 29 ........ ( 2 )

    ( 1 ) + ( 2 )

    6a = 6 , a = 1

    4(1) + b = 29 , b = 25

    (b). 2x³ + x² + 25x + 12 = 0

    [tex]x_{1}[/tex] = - 4 ( given: f(x) is divisible by x + 4 )

    [tex]x_{2}[/tex] = 0.5

    [tex]x_{3}[/tex] = 3