https://file.aiquickdraw.com/mathbot/file/1727358430714t1vor85c.jpg

We are given the polynomial \( P(x) = x^3 - x^2 + 4x - 30 \) and one of its roots, \( -1 + 3i \). Since the coefficients of the polynomial are real, the complex conjugate \( -1 - 3i \) must also be a root.

### Step-by-step solution:

1. Since \( -1 + 3i \) and \( -1 - 3i \) are roots, the quadratic factor associated with these roots is:
   \[
   (x - (-1 + 3i))(x - (-1 - 3i)) = (x + 1 - 3i)(x + 1 + 3i)
   \]
   This is a difference of squares:
   \[
   (x + 1)^2 - (3i)^2 = (x + 1)^2 + 9
   \]
   Expanding \( (x + 1)^2 \):
   \[
   (x + 1)^2 = x^2 + 2x + 1
   \]
   So, the quadratic factor becomes:
   \[
   x^2 + 2x + 1 + 9 = x^2 + 2x + 10
   \]

2. Now that we know one factor of \( P(x) \) is \( x^2 + 2x + 10 \), we can perform polynomial division to find the remaining factor. We will divide \( P(x) = x^3 - x^2 + 4x - 30 \) by \( x^2 + 2x + 10 \).

Let's carry out this division:

### Polynomial Division:

\[
\frac{x^3 - x^2 + 4x - 30}{x^2 + 2x + 10}
\]

- The first term of the quotient is \( x \), because \( x^3 / x^2 = x \).
- Multiply \( x \) by \( x^2 + 2x + 10 \):
  \[
  x(x^2 + 2x + 10) = x^3 + 2x^2 + 10x
  \]
- Subtract \( x^3 + 2x^2 + 10x \) from \( x^3 - x^2 + 4x - 30 \):
  \[
  (x^3 - x^2 + 4x - 30) - (x^3 + 2x^2 + 10x) = -3x^2 - 6x - 30
  \]

- The next term of the quotient is \( -3 \), because \( -3x^2 / x^2 = -3 \).
- Multiply \( -3 \) by \( x^2 + 2x + 10 \):
  \[
  -3(x^2 + 2x + 10) = -3x^2 - 6x - 30
  \]
- Subtract \( -3x^2 - 6x - 30 \) from \( -3x^2 - 6x - 30 \):
  \[
  (-3x^2 - 6x - 30) - (-3x^2 - 6x - 30) = 0
  \]

Thus, the quotient is \( x - 3 \), and there is no remainder.

### Conclusion:
The factorization of the polynomial is:
\[
P(x) = (x^2 + 2x + 10)(x - 3)
\]

### The roots are:
1. \( -1 + 3i \) (given),
2. \( -1 - 3i \) (the complex conjugate),
3. \( x = 3 \).

Would you like further details on the polynomial division or any part of this explanation?

---

Here are some related questions to expand your understanding:

1. How can you confirm that \( x^2 + 2x + 10 \) has no real roots?
2. What is the significance of complex conjugates when dealing with polynomials with real coefficients?
3. How would the factorization change if one of the roots was a real number?
4. Can you find the derivative of the polynomial and use it to determine critical points?
5. What is the behavior of the polynomial graph as \( x \to \infty \) and \( x \to -\infty \)?

**Tip:** Always remember that if a polynomial has real coefficients, complex roots come in conjugate pairs.

Find all other zeros of P(x) = x^3 - x^2 + 4x - 30, given that -1 + 3i is a zero.

This problem explores how to find the zeros of a cubic polynomial P(x) = x^3 - x^2 + 4x - 30, including complex conjugate roots. It applies the conjugate root theorem, polynomial factorization, and polynomial division to find the remaining real and complex roots.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation