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. In Exercises $53-56$, use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.
$$\frac{x^{2}-4 x+1}{x-i}$$

Key Concepts

Synthetic Division

Synthetic division is a streamlined process used to divide a polynomial by a linear factor of the form x - c. It simplifies the division process by focusing on the coefficients of the polynomial and performing arithmetic operations in a tabular form, thus avoiding the more cumbersome full polynomial long division. This method is efficient especially when one needs to quickly compute the quotient and remainder of the division.

Polynomial Division Algorithm

The polynomial division algorithm is analogous to the long division process for numbers. It guarantees that any dividend polynomial can be expressed as the product of the divisor and the quotient, plus a remainder that is of lower degree than the divisor. This concept underlines the fundamental division property for polynomials and is the theoretical basis for methods like synthetic division.

Complex Numbers in Polynomials

When polynomials involve complex coefficients or are divided by factors that lead to complex arithmetic, the operations still follow the same arithmetic rules. Handling complex coefficients requires an understanding of complex number arithmetic, ensuring that both the addition and multiplication of complex numbers are correctly executed during division. This demonstrates the versatility and universality of polynomial division procedures.

Remainder Theorem

The Remainder Theorem states that when a polynomial is divided by a linear divisor of the form x - c, the remainder is equivalent to evaluating the polynomial at x = c. This concept is especially useful because it offers a quick way to determine the remainder without executing the full division process, and it underpins the reasoning behind synthetic division.

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