Using the Given Zero, Find All Other Zeros of f(x) = x^4 + 45x^2 + 196

To find all the zeros of the polynomial function f(x) = x^4 + 45x^2 + 196, given that 2i is one of its zeros, we can start by recognizing that complex roots come in conjugate pairs. Therefore, if 2i is a zero, -2i must also be a zero.

Next, we can express the polynomial in terms of its roots. The zeros 2i and -2i can be represented as factors:

(x - 2i)(x + 2i) = x^2 + 4

This means that x^2 + 4 is a factor of f(x). Next, we can divide f(x) by x^2 + 4 to find the other factor. Performing polynomial long division:

x^4 + 45x^2 + 196 ÷ (x^2 + 4)

After completing the division, we find that:

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

Now we will set x^2 + 45 = 0 to find the remaining zeros:

x^2 = -45

Taking the square root gives us:

x = ±√(-45) = ±√(45)i = ±3√5 i

Thus, the complete set of zeros of the polynomial function f(x) includes:

• 2i
• -2i
• 3√5 i
• -3√5 i

In conclusion, the zeros of the polynomial function f(x) = x^4 + 45x^2 + 196 are 2i, -2i, 3√5 i, and -3√5 i.