If I know that 2i is a zero of the function f(x) = x^4 + 45x^2 + 196, how can I find all the other zeros?

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Finding All Zeros of the Polynomial

To find all the zeros of the polynomial function f(x) = x^4 + 45x^2 + 196, we start by recognizing that we already know one of the zeros is 2i (a complex number). Since the coefficients of our polynomial are real, the complex zeros will occur in conjugate pairs. This means that if 2i is a zero, then -2i is also a zero.

Step 1: Using the Known Zeros

We can factor out the zeros (x – 2i)(x + 2i) from the polynomial. This can be simplified to:

x^2 + 4

Thus, we can rewrite the polynomial:

f(x) = (x^2 + 4)(? ? ? ?)

Step 2: Dividing the Polynomial

Next, we will divide the original polynomial f(x) by x^2 + 4 using polynomial long division or synthetic division:


	1. Set up the division: f(x) ÷ (x^2 + 4)
	2. Perform the division to find the other quadratic factor.

Step 3: Finding the Remaining Zeros

After completing the polynomial division, we find another quadratic which would have the form ax^2 + bx + c.

For our polynomial f(x), we will ultimately find:

f(x) = (x^2 + 4)(x^2 + 45)

To find the zeros of the new factor x^2 + 45 = 0, we solve for:

x^2 = -45

This gives us:

x = ±√(-45) = ±√(45)i = ±3√5 i

Conclusion

In conclusion, the complete set of zeros for the polynomial function f(x) = x^4 + 45x^2 + 196 are:

  • 2i
  • -2i
  • 3√5 i
  • -3√5 i

These zeros include both real and complex solutions, providing a comprehensive picture of the function’s behavior.

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