Cn anyone solve these...?

the polynomial x^3 + ax^2 + bx - 3 leaves remainders of 27 when divided by x - 2 and a remainder of 3 when divided by x + 1. Calculate the remainder when the polynomial is divided by x - 1.

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    It is great that Yahoo provides a forum for help with basic algebra and it looks like Puggy already solved the problem.

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  • Puggy
    Lv 7
    1 decade ago

    Let p(x) = x^3 + ax^2 + bx - 3

    Since we get a remainder when dividing p(x) by (x - 2), it follows that p(2) = 27. However, p(2) is also equal to what is below:

    p(2) = 2^3 + a(2)^2 + b(2) - 3

    p(2) = 8 + 4a + 2b - 3

    p(2) = 5 + 4a + 2b

    So now we can equate this to 27, getting

    27 = 5 + 4a + 2b

    22 = 4a + 2b, and reducing this, we get

    11 = 2a + b

    We also know we get a remainder of 3 when dividing p(x) by (x + 1), so we know that p(-1) = 3. But, p(-1) is also equal to:

    p(-1) = (-1)^3 + a(-1)^2 + b(-1) - 3

    p(-1) = -1 + a - b - 3

    p(-1) = -4 + a - b

    Equating this to 3, we get

    3 = -4 + a - b

    7 = a - b

    Two equations, two unknowns:

    11 = 2a + b

    7 = a - b

    To solve this, we can use elimination and add the two equations.

    18 = 3a

    Therefore, a = 6

    Plugging this into the second equation,

    7 = 6 - b, therefore,

    1 = -b, so b = -1

    So a = 6 and b = -1

  • 1 decade ago

    By remainder theorem,

    2^3+a2^2+2b - 3 = 27

    which can be reduced to

    2a+b = 11......(1)

    (-1)^3+a-b - 3 = 3

    which can be reduced to

    a-b = 7......(2)

    Solving the system of (1) and (2) gives,

    a = 6, b = -1

    Therefore, the remainder when the polynomial is divided by x-1 is: 1+6-1-3 = 3

  • raj
    Lv 7
    1 decade ago

    using the remainder theorem

    f(2)=2^3+a(2)^2+b(2)-3=27

    so 4a+2b=22 (1)

    f(-1)=(-1)^3+a(-1)^2+b(-1)-3=3

    so a-b=7 (2)

    (2)*2

    2a-2b=14

    4a+2b=22

    adding

    6a=36

    dividing by 6

    a=6

    aub in (2)

    6-b=7

    -b=1

    b=-1

    so the poly is

    x^3+6x^2-b-3

    f(1) =1+6-1-3

    =3

    so the remainderwhen divided by x-1 is 3

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