Question

find the quotient and remainder using synthetic division: \\(\frac{x^3 + 2x^2 - 19}{x - 2}\\)
the quotient is
the remainder is
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Explanation:

Step1: Identify root & coefficients

Root of divisor $x-2$ is $2$.
Coefficients of dividend: $1$ (for $x^3$), $2$ (for $x^2$), $0$ (for $x$), $-19$ (constant)

Step2: Set up synthetic division

2 | 1   2   0  -19
        2   8   16
      ---------------
        1   4   8   -3

Step3: Extract quotient & remainder

Quotient uses first 3 results, degree reduced by 1.
Remainder is last result.

Answer:

The quotient is $x^2 + 4x + 8$
The remainder is $-3$