Question

find the quotient and remainder using synthetic division: \\(dfrac{x^4 - 3x^3 + 23x + 11}{x + 2}\\)
the quotient is \\(\boxed{1}\\)
the remainder is \\(\boxed{}\\)
question help: video read written example

Explanation:

Step1: Identify root of divisor

Divisor: $x+2=0 \implies x=-2$
Coefficients of dividend: $1, -3, 0, 23, 11$ (for $x^4, x^3, x^2, x, \text{constant}$)

Step2: Set up synthetic division

Bring down the leading coefficient:
$1$
Multiply by $-2$: $1 \times (-2) = -2$
Add to next coefficient: $-3 + (-2) = -5$

Step3: Continue synthetic division

Multiply $-5$ by $-2$: $10$
Add to next coefficient: $0 + 10 = 10$

Step4: Next synthetic division step

Multiply $10$ by $-2$: $-20$
Add to next coefficient: $23 + (-20) = 3$

Step5: Final synthetic division step

Multiply $3$ by $-2$: $-6$
Add to last coefficient: $11 + (-6) = 5$

Step6: Write quotient and remainder

Coefficients of quotient: $1, -5, 10, 3$ (degree 3 polynomial)
Remainder: $5$

Answer:

The quotient is $x^3 - 5x^2 + 10x + 3$
The remainder is $5$