Question

use synthetic division to simplify \\(\frac{x^2 - 3x}{x - 3}\\). write your answer in the form \\(q(x) + \frac{r}{d(x)}\\), where \\(q(x)\\) is a polynomial, \\(r\\) is an integer, and \\(d(x)\\) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Identify the root of the divisor

The divisor is \(x - 3\), so the root \(c = 3\). The coefficients of the dividend \(x^{2}-3x\) are \(1\) (for \(x^{2}\)), \(-3\) (for \(x\)), and \(0\) (constant term, since there's no constant term).

Step2: Set up synthetic division

Write the root \(3\) to the left, and the coefficients \(1\), \(-3\), \(0\) in a row:
\[

$$\begin{array}{r|rrr} 3 & 1 & -3 & 0 \\ & & 3 & 0 \\ \hline & 1 & 0 & 0 \\ \end{array}$$

\]
We bring down the first coefficient \(1\). Then multiply \(3\) by \(1\) to get \(3\), add to \(-3\) to get \(0\). Then multiply \(3\) by \(0\) to get \(0\), add to \(0\) to get \(0\).

Step3: Interpret the result

The quotient polynomial \(q(x)\) has coefficients \(1\) (for \(x\)) and \(0\) (constant term), so \(q(x)=x + 0=x\). The remainder \(r = 0\), and the divisor \(d(x)=x - 3\). So the expression simplifies to \(x+\frac{0}{x - 3}\), which is just \(x\) (since \(\frac{0}{x - 3}=0\)).

Answer:

\(x+\frac{0}{x - 3}\) (or simply \(x\) as the fraction term is zero)