The process of synthetic division applies equally well when...

The process of synthetic division applies equally well when some or all of the coefficients are nonreal complex numbers. Use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.
$$\frac{x^{3}-2 x^{2}-4}{x-3 i}$$

Key Concepts

Remainder Theorem

The Remainder Theorem asserts that when a polynomial is divided by a binomial of the form (x ? c), the remainder is equal to the value of the polynomial evaluated at c. This concept holds true regardless of whether c is a real or a complex number, providing a quick check for the outcome of a division and reinforcing the connection between polynomial division and function evaluation.

Complex Numbers

Complex numbers extend the real number system by including an imaginary unit i, where i² = -1. When working with polynomials that have complex coefficients or when the divisor involves complex numbers, the same arithmetic rules apply. However, one must carefully handle addition, multiplication, and powers of i during the division process.

Polynomial Division Algorithm

The polynomial division algorithm states that any polynomial can be expressed as the product of a divisor and a quotient plus a remainder. This general framework provides a systematic approach to breaking down polynomial expressions, ensuring that the degree of the remainder is lower than the degree of the divisor, whether the coefficients are real or complex.

Synthetic Division

Synthetic division is a streamlined method used for dividing a polynomial by a binomial of the form (x ? c). It simplifies the division process by only working with the coefficients of the polynomial and is efficient for calculating both the quotient and the remainder. This method is valid not only for real numbers but also for complex numbers, making it a versatile tool in algebra.

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