How can we find the complex zeros of the polynomial function and express it in its factored form?

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Finding the Complex Zeros of a Polynomial Function

To find the complex zeros of a polynomial function, we typically follow these steps:

  1. Write Down the Polynomial:

    First, identify the polynomial function you want to analyze. For example, let’s consider the polynomial function f(x) = x3 + 2x2 + 5x + 6.

  2. Use the Rational Root Theorem:

    Check for possible rational roots using the Rational Root Theorem. This involves testing potential rational zeros, which are factors of the constant term divided by factors of the leading coefficient.

  3. Perform Synthetic Division:

    If you find any rational roots, perform synthetic division or polynomial long division to simplify the polynomial. This will help you reduce the polynomial degree to find remaining roots.

  4. Identify Remaining Roots:

    The roots can be either real or complex. If any quadratic factor remains after decomposition, apply the quadratic formula:

    x = (-b ± √(b² – 4ac)) / 2a

  5. Look for Complex Zeros:

    If the discriminant (b² – 4ac) is negative, this indicates the presence of complex zeros. For example, if you find (x – 1)(x2 + 2), you will proceed to solve the quadratic part.

  6. Express in Factored Form:

    Finally, express the polynomial in factored form. If the complex zeros were found, you might write them as:

    f(x) = (x – z1)(x – z2)(x – z3), where z1, z2, and z3 include both real and complex roots.

Example

Let’s work through an example:

Given f(x) = x3 + 2x2 + 5x + 6:

  1. Testing for rational roots, we find that x = -1 is a root.
  2. Now, using synthetic division, we divide f(x) by (x + 1), resulting in:

    3. f(x) = (x + 1)(x2 + x + 6)

  3. Next, apply the quadratic formula to the quadratic component (x2 + x + 6). This will yield complex roots.
  4. Thus, the final factored form is:

    5. f(x) = (x + 1)(x – (-1 + √23i))(x – (-1 – √23i))

In summary, finding complex zeros requires a systematic approach including testing for rational roots, polynomial division, and the use of the quadratic formula to identify complex solutions. Finally, represent the polynomial function in its complete factored form.

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