Question

1. $(x^3 - 2x + 1)/(x - 1)$
2. $x^2 + x - 1$
3. $(2x^4 + 2x^3 - 12x^2 + x + 6)/(x + 3)$
4. $2x^3 - 4x^2 + 1 + 3/(x + 3)$

Explanation:

Step1: Set up synthetic division for Q3

Use root $x=1$, coefficients: $1, 0, -2, 1$

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

Step2: Interpret synthetic division result

Quotient coefficients: $1,1,-1$, remainder $0$

Step3: Set up synthetic division for Q4

Use root $x=-3$, coefficients: $2, 2, -12, 1, 6$

$$\begin{array}{r|rrrrr} -3 & 2 & 2 & -12 & 1 & 6 \\ & & -6 & 12 & 0 & -3 \\ \hline & 2 & -4 & 0 & 1 & 3 \\ \end{array}$$

Step4: Interpret synthetic division result

Quotient coefficients: $2,-4,0,1$, remainder $3$

Answer:

1. $\boldsymbol{x^2 + x - 1}$
2. $\boldsymbol{2x^3 - 4x^2 + 1 + \frac{3}{x+3}}$