Given a zero of the polynomial, determine all other zeros...

Given a zero of the polynomial, determine all other zeros (real and complex) and write the polynomial in terms of a product of linear factors.
Zero
$-2 i$
$i$
$-3 i$
$1+i$
$3-i$
$2-i$
$3 i$
$-2 i$
$5 i$
$1-2 i$
$2+i$
$3+i$
$$P(x)=x^{4}-9 x^{3}+29 x^{2}-41 x+20$$

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Key Concepts

Fundamental Theorem of Algebra

This theorem guarantees that any non?zero single-variable polynomial with complex coefficients has exactly as many roots (counting multiplicities) as its degree. In other words, a degree 4 polynomial will have four roots in the complex number system, a fact that is essential when determining all zeros of a polynomial.

Complex Conjugate Root Theorem

For polynomials with real coefficients, any non-real complex zero must appear together with its complex conjugate. This means that if a+bi is a zero of the polynomial, then its conjugate a-bi is also a zero. This theorem is crucial when factoring polynomials over the reals, ensuring that complex zeros occur in pairs.

Factor Theorem

The Factor Theorem states that if a polynomial f(x) has a zero at x = a, then (x - a) is a factor of the polynomial. This provides a direct method for factorizing the polynomial by relating its zeros to its linear factors.

Polynomial Factorization into Linear Factors

This concept involves expressing a polynomial as a product of linear factors of the form (x - r), where each r is a root of the polynomial. It is particularly useful for solving polynomial equations and analyzing the polynomial's behavior, since each factor corresponds directly to one of its zeros.

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