Question

use synthetic division to find the result when $x^3 + 3x^2 - x + 4$ is divided by $x - 1$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Set up synthetic division

For dividing \(x^{3}+3x^{2}-x + 4\) by \(x - 1\), we use the root of the divisor \(x-1=0\) i.e., \(x = 1\). The coefficients of the dividend are \(1,3,-1,4\).
\[

$$\begin{array}{r|rrrr} 1 & 1 & 3 & -1 & 4\\ & & 1 & 4 & 3\\ \hline & 1 & 4 & 3 & 7\\ \end{array}$$

\]

Step2: Interpret the results

The last number \(7\) is the remainder. The other numbers \(1,4,3\) are the coefficients of the quotient polynomial. So the quotient \(q(x)=x^{2}+4x + 3\) and the remainder \(r(x)=7\), the divisor \(b(x)=x - 1\).

Step3: Write in the required form

The result is \(q(x)+\frac{r(x)}{b(x)}=x^{2}+4x + 3+\frac{7}{x - 1}\)

Answer:

\(x^{2}+4x + 3+\frac{7}{x - 1}\)