Given P(x)=P(x)=3 x^5 - 5 x^4 + 64 x^3 - 128 x^2 - 275 x - 75, and that 5i is a zero, write P in factored form

Given \( P(x) = 3x^5 - 5x^4 + 64x^3 - 128x^2 - 275x - 75 \) and the fact that \( 5i \) is a zero, we can use the fact that if \( 5i \) is a zero, its conjugate \( -5i \) must also be a zero since the coefficients of the polynomial are real.

### Step-by-Step Process:

1. **Use the known zeros \( 5i \) and \( -5i \)**:
   - The quadratic factor corresponding to these two complex zeros is:
     \[
     (x - 5i)(x + 5i) = x^2 + 25
     \]
2. **Divide the polynomial \( P(x) \) by \( x^2 + 25 \)**:
   - We now divide the given polynomial by \( x^2 + 25 \) to factor it further.

Let's perform the division now:

\[
P(x) = 3x^5 - 5x^4 + 64x^3 - 128x^2 - 275x - 75
\]

Dividing \( P(x) \) by \( x^2 + 25 \):

The division of \( P(x) \) by \( x^2 + 25 \) gives the quotient \( 3x^3 - 5x^2 - 11x - 3 \) with no remainder, meaning the factorization process is correct.

### Factored Form of \( P(x) \):
\[
P(x) = (x^2 + 25)(3x^3 - 5x^2 - 11x - 3)
\]

This is the factored form of the polynomial \( P(x) \).

Would you like more details on the process or have any specific questions?

Here are some related questions:
1. How do you factor a cubic polynomial?
2. What are complex conjugates and why are they important in polynomial factorization?
3. How can synthetic division be used in polynomial division?
4. What is the significance of the remainder in polynomial division?
5. How do the degrees of polynomials influence the factoring process?

**Tip:** When factoring polynomials with complex roots, always consider their conjugates, as real polynomials must have real coefficients.

Given P(x)= 3x^5 - 5x^4 + 64x^3 - 128x^2 - 275x - 75, and that 5i is a zero, write P in factored form.

Learn how to factor the polynomial P(x) = 3x^5 - 5x^4 + 64x^3 - 128x^2 - 275x - 75 given that 5i is a zero. This problem explores polynomial factorization, complex conjugates, and polynomial division. Suitable for high school students in Grades 10-12.

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