Question

find the quotient and remainder using synthetic division for: $\frac{x^{3}+6x^{2}+12x+15}{x+2}$
the quotient is
the remainder is
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Explanation:

Step1: Identify root of divisor

The divisor is $x+2$, so solve $x+2=0$ to get $x=-2$.

Step2: List coefficients of dividend

Dividend: $x^3+6x^2+12x+15$, coefficients are $1, 6, 12, 15$.

Step3: Set up synthetic division

Bring down the leading coefficient: $1$
Multiply by $-2$: $1\times(-2)=-2$
Add to next coefficient: $6+(-2)=4$

Step4: Continue synthetic division

Multiply $4$ by $-2$: $4\times(-2)=-8$
Add to next coefficient: $12+(-8)=4$

Step5: Final synthetic division step

Multiply $4$ by $-2$: $4\times(-2)=-8$
Add to last coefficient: $15+(-8)=7$

Step6: Form quotient from coefficients

The first three results $1, 4, 4$ are coefficients of the quotient polynomial of degree $2$: $x^2+4x+4$
The last result is the remainder.

Answer:

The quotient is $x^2+4x+4$
The remainder is $7$