Question

which shows the correct division? \\((3x^4 - 8x^2 - x + 1)(x + 2)^{-1}\\) (use synthetic division) \\(3x^2 - 14x + 27 - \frac{53}{x + 2}\\) \\(3x^3 + 6x^2 + 4x + 7 + \frac{15}{x + 2}\\) \\(3x^3 - 6x^2 + 4x - 9 + \frac{19}{x + 2}\\) \\(3x^2 - 2x - 5 - \frac{9}{x + 2}\\)

Explanation:

Step1: Set up synthetic division

Root for divisor $x+2$ is $-2$. Coefficients of dividend $3x^4 - 8x^2 - x + 1$: $3, 0, -8, -1, 1$ (add 0 for missing $x^3$ term).

Step2: Perform synthetic division

Bring down 3. Multiply by -2: -6. Add to next coefficient: $0 + (-6) = -6$.
Multiply -6 by -2: 12. Add to next coefficient: $-8 + 12 = 4$.
Multiply 4 by -2: -8. Add to next coefficient: $-1 + (-8) = -9$.
Multiply -9 by -2: 18. Add to last coefficient: $1 + 18 = 19$.

Step3: Write quotient and remainder

Quotient: $3x^3 - 6x^2 + 4x - 9$. Remainder: 19.
Result: $3x^3 - 6x^2 + 4x - 9 + \frac{19}{x+2}$

Answer:

3x³ - 6x² + 4x - 9 + $\frac{19}{x+2}$