given the function f(x)=x^4 +7x^2 -18 and that 3i is a root. What are the factors of f(x)?

Given the polynomial function \( f(x) = x^4 + 7x^2 - 18 \) and knowing that \( 3i \) is a root, we can also conclude that \( -3i \) is a root due to the complex conjugate root theorem.

### Step 1: Formulate Factors from Roots
The roots \( 3i \) and \( -3i \) correspond to the quadratic factor:
\[
(x - 3i)(x + 3i) = x^2 + 9
\]

### Step 2: Divide \( f(x) \) by \( x^2 + 9 \)
To find the other factor, perform polynomial long division of \( f(x) \) by \( x^2 + 9 \).

1. Divide \( x^4 \) by \( x^2 \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( x^2 + 9 \):
   \[
   x^2(x^2 + 9) = x^4 + 9x^2
   \]
3. Subtract from \( f(x) \):
   \[
   (x^4 + 7x^2 - 18) - (x^4 + 9x^2) = -2x^2 - 18
   \]
4. Divide \( -2x^2 \) by \( x^2 \) to get \( -2 \).
5. Multiply \( -2 \) by \( x^2 + 9 \):
   \[
   -2(x^2 + 9) = -2x^2 - 18
   \]
6. Subtract again:
   \[
   (-2x^2 - 18) - (-2x^2 - 18) = 0
   \]

### Step 3: Final Factors
Thus, we have:
\[
f(x) = (x^2 + 9)(x^2 - 2)
\]

### Summary of Factors
The complete factorization of \( f(x) \) is:
\[
f(x) = (x^2 + 9)(x^2 - 2)
\]

### Further Breakdown of Factors
The quadratic \( x^2 - 2 \) can be factored further:
\[
x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2})
\]

### Final Factorization
Therefore, the complete factorization of \( f(x) \) is:
\[
f(x) = (x - \sqrt{2})(x + \sqrt{2})(x^2 + 9)
\]

Would you like more details on any of these steps? Here are some related questions to consider:

1. How do you verify if a polynomial has a specific root?
2. What are the implications of complex roots in polynomial equations?
3. How can you graph the function to visualize the roots?
4. What is the role of the degree of the polynomial in determining the number of roots?
5. How can synthetic division be used as an alternative method for polynomial division?

**Tip:** Always check if the roots are real or complex before attempting factorization.

Given the function f(x)=x^4 +7x^2 -18 and that 3i is a root. What are the factors of f(x)?

Discover the step-by-step process for factoring the polynomial f(x) = x^4 + 7x^2 - 18, knowing that 3i is a root. This guide covers key mathematical concepts such as complex roots and polynomial long division, suitable for advanced high school students.

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