1.3 Complex Polynomials as a Product of Linear Factors

1.3 Complex Polynomials as a Product of Linear Factors

A complex polynomial $P(z)$ is a polynomial function of the complex variable $z$ with complex coefficients. It is expressed in the general form

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are complex numbers ($a_n \neq 0$), and $n \ge 0$ is the degree of the polynomial.

For example:

• $P_1(z) = (1 - i)z + 3i$ (linear)
• $P_2(z) = (5 - 4i)z^2 + (2 + i)z + (3 - 4i)$ (quadratic)
• $P_3(z) = (2 - i)z^3 + 2z^2 i + (5 + 3i)$ (cubic)

If $n = 0$, then $P(z)$ is a constant polynomial.

A fundamental property of complex polynomials is that they can always be factored into a product of linear factors.

According to the Fundamental Theorem of Algebra, a polynomial of degree $n \ge 1$ has exactly $n$ roots in the complex number system $\mathbb{C}$.

A corollary states that any polynomial $P(z)$ of degree $n$ can be factored completely into a constant $a$ and $n$ linear factors over $\mathbb{C}$:

where $z_1, z_2, \ldots, z_n$ are the complex roots of $P(z) = 0$. Once the roots are known, equation (1) gives the factorization into linear factors.

For example, $P(x) = x^2 + 4$ has no real roots, so it cannot be factored into linear polynomials with real coefficients. As a complex polynomial $P(z) = z^2 + 4$:

where $2i$ and $-2i$ are the complex roots of $z^2 + 4 = 0$.

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Note. If $P(z)$ is a polynomial function, the values of $z$ that satisfy $P(z) = 0$ are called the zeros of $P(z)$ and the roots of the equation $P(z) = 0$.

Example 6

Factorize the polynomial $P(z) = z^2 + (i - 3)z - 3i$.

Solution

Example 7

Factorize the polynomial $P(z) = z^2 - 4iz + 12$.

Solution

Example 8

Factorize the polynomial $P(z) = z^3 + (1 + i)z^2 + iz$.

Solution