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Math Help - find polynomial f(x)

  1. #1
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    find polynomial f(x)

    I have a test up tomorrow and I really need to know how to solve this; somehow I didn't write explanations in my notes so although I kind of have a vague idea of how to solve it, I don't get it.

    Find a polynomial f(x) of degree 4, given that f(1)=f(2)=f(-2)=0 and f(3)=f(-3)=40.

    thank you so much in advance.
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    Quote Originally Posted by colloquial View Post
    I have a test up tomorrow and I really need to know how to solve this; somehow I didn't write explanations in my notes so although I kind of have a vague idea of how to solve it, I don't get it.

    Find a polynomial f(x) of degree 4, given that f(1)=f(2)=f(-2)=0 and f(3)=f(-3)=40.

    thank you so much in advance.
    Well, we know that 1, 2, and -2 are zeros. So the 4th zero is real as well. (If it were complex then there would be a fifth zero: the complex conjugate of the fourth zero.) So the polynomial is of the form:
    f(x) = (x - 1)(x - 2)(x + 2)(x - a)
    where a is the fourth zero, currently unknown.

    f(x) = x^4 - (a + 1)x^3 +(a - 4)x^2 + (4a +4)x - 4a

    We also know that
    f(3)= f(-3) = 40

    From our polynomial we know that
    f(3) = -10a + 30 = 40
    so a = -1.

    (The condition f(-3) = 40 gives us the same value.)

    Thus
    f(x) = x^4 - 5x^2 + 4

    -Dan
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by colloquial View Post
    I have a test up tomorrow and I really need to know how to solve this; somehow I didn't write explanations in my notes so although I kind of have a vague idea of how to solve it, I don't get it.

    Find a polynomial f(x) of degree 4, given that f(1)=f(2)=f(-2)=0 and f(3)=f(-3)=40.

    thank you so much in advance.
    Here's another approach, a much longer one. Set up a simultaneous equation system of five equations and five unknowns.

    Let the polynomial be of the form: ax^4 + bx^3 + cx^2 + dx + e

    Now,

    f(1) = a + b + c + d + e = 0 ..........................(1)

    f(2) = 16a + 8b + 4c + 2d + e = 0 ..................(2)

    f(-2) = 16a - 8b + 4c - 2d + e = 0 ................(3)

    f(3) = 81a + 27b + 9c + 3d + e = 40 ...............(4)

    f(-3) = 81a - 27b + 9c -3d + e = 40 ..............(5)

    Now solve the above system to get your coefficients
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