Factor the function p(x)=3x5-5x4-13x3-65x2-2200x+1500 over the complex number system. Given: -5i is a zero.

To factor the polynomial \( p(x) = 3x^5 - 5x^4 - 13x^3 - 65x^2 - 2200x + 1500 \) over the complex number system, given that \( -5i \) is a zero, we can use the fact that for real-coefficient polynomials, complex roots always come in conjugate pairs. Therefore, if \( -5i \) is a zero, its conjugate \( 5i \) is also a zero.

We will proceed as follows:

### Step 1: Use the known roots to create a factor.
Since \( -5i \) and \( 5i \) are both roots, \( (x + 5i)(x - 5i) = x^2 + 25 \) is a factor of the polynomial.

### Step 2: Perform synthetic or long division of \( p(x) \) by \( x^2 + 25 \).

Let's divide the polynomial by \( x^2 + 25 \) to reduce the degree.

\[
\frac{3x^5 - 5x^4 - 13x^3 - 65x^2 - 2200x + 1500}{x^2 + 25}
\]

I'll now calculate this division.

### Step 3: Use the quotient to factor further.

The division of \( p(x) \) by \( x^2 + 25 \) gives:

\[
\frac{p(x)}{x^2 + 25} = 3x^3 - 5x^2 - 88x + 60
\]

Since the remainder is 0, \( p(x) \) can be factored as:

\[
p(x) = (x^2 + 25)(3x^3 - 5x^2 - 88x + 60)
\]

### Step 4: Factor the cubic polynomial \( 3x^3 - 5x^2 - 88x + 60 \).

We'll now attempt to factor the cubic polynomial further.

The cubic polynomial \( 3x^3 - 5x^2 - 88x + 60 \) factors as:

\[
3x^3 - 5x^2 - 88x + 60 = (x - 6)(x + 5)(3x - 2)
\]

### Step 5: Final Factorization

Thus, the full factorization of \( p(x) = 3x^5 - 5x^4 - 13x^3 - 65x^2 - 2200x + 1500 \) over the complex number system is:

\[
p(x) = (x^2 + 25)(x - 6)(x + 5)(3x - 2)
\]

This is the complete factorization of the polynomial.

Would you like more details on any step or have any further questions?

Here are five related questions to expand this topic:
1. How do you use synthetic division to factor polynomials?
2. Why do complex roots of real polynomials always come in conjugate pairs?
3. Can every cubic polynomial be factored into linear factors?
4. How would you solve for all the roots of the factored polynomial?
5. What is the significance of factoring a polynomial in terms of its roots?

**Tip:** When dividing polynomials, check if the remainder is zero to confirm that a factorization is successful.

Factor the function p(x) = 3x^5 - 5x^4 - 13x^3 - 65x^2 - 2200x + 1500 over the complex number system. Given: -5i is a zero.

Learn how to factor the polynomial 3x^5 - 5x^4 - 13x^3 - 65x^2 - 2200x + 1500 given that -5i is a zero. This problem explores polynomial factorization, complex roots, and synthetic division. Ideal for students in Grades 11-12 and college level.

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