Given the degree of the polynomial $f(x)$ is 4 and you are provided with two zeros: $i$ and $\sqrt{2}$, let's find the remaining zeros.

Step 1: Use the fact that complex roots come in conjugate pairs

Since $i$ is a zero, its complex conjugate $-i$ must also be a zero. So, two of the zeros are $i$ and $-i$.

Step 2: Write down the known zeros

We are now given:

• $i$
• $-i$
• $\sqrt{2}$

So far, we have three zeros. Since the polynomial is of degree 4, we need one more zero.

Step 3: Use the fact that irrational roots also come in conjugate pairs

If $\sqrt{2}$ is a zero, its conjugate $-\sqrt{2}$ must also be a zero. Therefore, the fourth zero is $-\sqrt{2}$.

Step 4: List all the zeros

The zeros of the polynomial $f(x)$ are:

1. $i$
2. $-i$
3. $\sqrt{2}$
4. $-\sqrt{2}$

These are all the zeros of the degree-4 polynomial.

Would you like more details on constructing the polynomial from these zeros or have any other questions?

Here are 5 related questions to expand on this:

1. How do you construct a polynomial from its zeros?
2. Why do complex roots come in conjugate pairs for polynomials with real coefficients?
3. What is the relationship between the degree of a polynomial and the number of its zeros?
4. How can you determine if a polynomial has imaginary or irrational roots?
5. How can you factor a polynomial given its roots?

Tip: When dealing with polynomials, always check if the coefficients are real. This often helps determine whether complex or irrational roots come in pairs.