In questions, two polynomials P and D are given. Use either...

In questions, two polynomials $P$ and $D$ are given. Use either synthetic division or long division to divide $P(x)$ by $D(x)$, and express $P(x)$ in the form $P(x)=D(x) \cdot Q(x)+R(x)$.
$$P(x)=3 x^{4}-8 x^{3}+9 x+5, D(x)=x-2$$

Key Concepts

Long Division of Polynomials

Long division of polynomials is a systematic method that generalizes the familiar long division algorithm for numbers. It involves dividing the highest degree term of the dividend by the highest degree term of the divisor to obtain a term in the quotient, multiplying the entire divisor by that term, subtracting the result from the dividend, and repeating the process until the remainder is of lower degree than the divisor.

Synthetic Division

Synthetic division is a simplified form of polynomial division that is particularly efficient when dividing by a linear factor of the form x - c. It streamlines the process by focusing only on the coefficients, reducing the amount of written computation, and it can be used to quickly determine the quotient and remainder.

Polynomial Division

Polynomial division is a process analogous to numerical long division, where one polynomial (the dividend) is divided by another (the divisor). The objective is to express the dividend as the product of the divisor and a quotient polynomial, plus a remainder polynomial whose degree is less than that of the divisor.

Quotient and Remainder

In polynomial division, the result is expressed in the form P(x) = D(x) · Q(x) + R(x), where Q(x) is the quotient polynomial and R(x) is the remainder polynomial. The degree of R(x) is always less than the degree of the divisor D(x), ensuring that the division is complete.

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