Question

use synthetic division to find $(x^{2} + 3x - 10) \div (x - 2)$. 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

For the divisor \(x - 2\), set \(x - 2 = 0\), so \(x = 2\). We will use 2 in synthetic division.

Step2: Set up synthetic division

The coefficients of the dividend \(x^{2}+3x - 10\) are 1 (for \(x^{2}\)), 3 (for \(x\)), and - 10 (constant term). So we set up the synthetic division as:
\[

$$\begin{array}{r|rrr} 2 & 1 & 3 & - 10\\ & & 2 & 10\\ \hline & 1 & 5 & 0\\ \end{array}$$

\]
The first number 1 (below the line) is the coefficient of \(x\) in the quotient, 5 is the constant term of the quotient, and 0 is the remainder.

Step3: Write the quotient and remainder

The quotient \(q(x)\) is \(x + 5\) (since the coefficients are 1 for \(x\) and 5 for the constant term) and the remainder \(r = 0\), and the divisor \(d(x)=x - 2\). So the result is \(q(x)+\frac{r}{d(x)}=(x + 5)+\frac{0}{x - 2}\) or simply \(x + 5\) (since \(\frac{0}{x - 2}=0\)).

Answer:

\(x + 5+\frac{0}{x - 2}\) (or \(x + 5\) as the remainder is 0)