Question

22.
2 | 1 -5 3
2 -6
1 -3 -3
\frac{x^{3} - 5x + 3}{x - 2} = x^{2} - 3x - \frac{3}{x - 2}

Explanation:

Step1: Identify polynomial division error

The dividend is $x^3 + 0x^2 -5x +3$, so synthetic division must use coefficients $1, 0, -5, 3$ (not $1, -5, 3$, which omits the $x^2$ term coefficient).

Step2: Perform correct synthetic division

Divide by $x-2$, so use root $2$:

$$\begin{array}{r|rrrr} 2 & 1 & 0 & -5 & 3 \\ & & 2 & 4 & -2 \\ \hline & 1 & 2 & -1 & 1 \end{array}$$

Step3: Rewrite the division result

The quotient is $x^2 + 2x -1$, remainder is $1$. So:
$\frac{x^3 -5x +3}{x-2} = x^2 + 2x -1 + \frac{1}{x-2}$

Answer:

The original work omitted the coefficient of $x^2$ (which is 0) in the dividend. The correct result is $\boldsymbol{\frac{x^3 -5x +3}{x-2} = x^2 + 2x -1 + \frac{1}{x-2}}$