Find the remaining zeros of f using the given information about the polynomial. Degree 4; zeros: I, 2^1/2

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.

Find the remaining zeros of f using the given information about the polynomial. Degree 4; zeros: i, 2^1/2

This solution helps find the remaining zeros of a degree 4 polynomial given two zeros: i and √2. It covers key algebraic concepts, including complex and irrational conjugates, and is suitable for high school students in Grades 10-12.

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