Find the complex zeros of the following polynomial function. Write f in factored form.
f left parenthesis x right parenthesis equals x cubed minus 12 x squared plus 49 x minus 58


We are given the polynomial function:

\[
f(x) = x^3 - 12x^2 + 49x - 58
\]

To find the complex zeros of the polynomial, we will use methods such as synthetic division, factoring, and the quadratic formula if necessary.

### Step 1: Apply the Rational Root Theorem

The Rational Root Theorem suggests that possible rational roots are factors of the constant term \((-58)\) divided by factors of the leading coefficient \((1)\). 

Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 29, \pm 58 \).

### Step 2: Test Possible Rational Roots

Let's test \(x = 2\) using synthetic division.

#### Synthetic Division by \(x = 2\):

\[
\begin{array}{r|rrrr}
2 & 1 & -12 & 49 & -58 \\
  &   & 2  & -20 & 58  \\
\hline
  & 1 & -10 & 29  & 0 \\
\end{array}
\]

Since the remainder is 0, \(x = 2\) is a root. Thus, we can factor \(f(x)\) as:

\[
f(x) = (x - 2)(x^2 - 10x + 29)
\]

### Step 3: Solve the Quadratic \(x^2 - 10x + 29 = 0\)

Now we solve the quadratic \(x^2 - 10x + 29 = 0\) using the quadratic formula:

\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(29)}}{2(1)}
\]

\[
x = \frac{10 \pm \sqrt{100 - 116}}{2}
\]

\[
x = \frac{10 \pm \sqrt{-16}}{2}
\]

\[
x = \frac{10 \pm 4i}{2}
\]

\[
x = 5 \pm 2i
\]

Thus, the remaining two roots are \(x = 5 + 2i\) and \(x = 5 - 2i\).

### Step 4: Write the Polynomial in Factored Form

The factored form of the polynomial is:

\[
f(x) = (x - 2)(x - (5 + 2i))(x - (5 - 2i))
\]

### Summary of Complex Zeros:

The complex zeros are:
- \(x = 2\)
- \(x = 5 + 2i\)
- \(x = 5 - 2i\)

Would you like more details on any step? 

Here are 5 related questions:
1. How does the Rational Root Theorem work?
2. What is synthetic division, and how is it used?
3. Can all polynomials be factored using real numbers?
4. How do complex conjugates affect polynomial factorization?
5. How do imaginary numbers arise from negative discriminants?

**Tip**: When dealing with complex roots, they always appear in conjugate pairs (e.g., \(a + bi\) and \(a - bi\)) if the coefficients of the polynomial are real.

Find the complex zeros of the following polynomial function. Write f in factored form. f(x) = x^3 - 12x^2 + 49x - 58

Learn how to find the complex zeros and write the factored form of the polynomial f(x) = x^3 - 12x^2 + 49x - 58. This problem covers algebraic concepts such as the Rational Root Theorem, synthetic division, and the quadratic formula. Suitable for students in Grades 10-12.

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