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Section A.1 Polynomial Division
Definition A.1 .
Let \(P(x)\) and \(D(x)\) be polynomials with \(D(x)\neq 0\text{.}\) The process of finding polynomials \(Q(x)\) and \(R(x)\) such that
\begin{equation*}
P(x) = D(x)Q(x) + R(x), \qquad \deg(R) \lt \deg(D)
\end{equation*}
is called polynomial division . In this identity, \(P(x)\) is the dividend , \(D(x)\) is the divisor , \(Q(x)\) is the quotient , and \(R(x)\) is the remainder .
The step-by-step alignment method is called
polynomial long division .
Example A.2 .
Divide
\(2x^3 + 3x^2 - 5x + 6\) by
\(x - 2\text{.}\)
The leading term of the dividend is \(2x^3\text{.}\) Since \(2x^3/(x)=2x^2\text{,}\) place \(2x^2\) in the quotient. Multiply and subtract:
\begin{equation*}
(2x^3 + 3x^2 - 5x + 6) - (2x^3 - 4x^2) = 7x^2 - 5x + 6.
\end{equation*}
Now divide \(7x^2\) by \(x\text{,}\) giving \(7x\text{.}\) Add \(7x\) to the quotient, then multiply and subtract:
\begin{equation*}
(7x^2 - 5x + 6) - (7x^2 - 14x) = 9x + 6.
\end{equation*}
Next divide \(9x\) by \(x\text{,}\) giving \(9\text{.}\) Add \(9\) to the quotient, then multiply and subtract:
\begin{equation*}
(9x + 6) - (9x - 18) = 24.
\end{equation*}
Therefore,
\begin{equation*}
2x^3 + 3x^2 - 5x + 6 = (x-2)(2x^2 + 7x + 9) + 24.
\end{equation*}
So the quotient is \(2x^2 + 7x + 9\) and the remainder is \(24\text{.}\)
