How can I find all the zeros of the polynomial function f(x) = x^4 – 32x^2 + 144 given that 2i is one of its zeros?

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To find all the zeros of the polynomial function f(x) = x^4 – 32x^2 + 144 given that 2i is a zero, we can follow these steps:

  1. Use the Given Zero: Since 2i is a zero, its complex conjugate, -2i, must also be a zero because the coefficients of the polynomial are real.
  2. Factor the Polynomial: We can express the zeros in terms of factors. The known zeros allow us to factor the polynomial as follows:
    f(x) = (x – 2i)(x + 2i)(g(x)), where g(x) is a quadratic polynomial.
  3. Simplify the Factors: The factorization of the complex zeros gives us:
    (x – 2i)(x + 2i) = x^2 + 4
  4. Divide the Polynomial: Now, we need to perform polynomial long division or synthetic division to divide f(x) by x^2 + 4.
    After dividing, we can find the quotient which should also be a quadratic polynomial:
    g(x) = x^2 – 32
  5. Find the Zeros of g(x): The quadratic g(x) = x^2 – 32 can be solved by setting it equal to zero:
    x^2 – 32 = 0
    Solving for x gives us:
    x^2 = 32
    x = ±√32 = ±4√2

Now, we have all the zeros of the polynomial f(x):

  • 2i
  • -2i
  • 4√2
  • -4√2

In summary, the complete set of zeros for the polynomial function f(x) = x^4 – 32x^2 + 144 are:

  • 2i
  • -2i
  • 4√2
  • -4√2

This approach highlights the connection between the zeros of the function and the factorization of the polynomial, which is a fundamental concept in algebra.

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