Use the remainder theorem to find $$ P(2) $$ for $$ P(x)=x^{4}-3 x^{3}-6 x^{2}+6 $$. Specifically, give the quotient and the remainder for the associated division and the value of $$ P(2) $$. Quotient $$ =\square $$ Remainder $$ =\square $$

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1. Write the polynomial with all coefficients, including the missing term for $$ x $$:
   $$
   P(x)=x^4 - 3x^3 - 6x^2 + 0x + 6.
   $$

2. To divide $$ P(x) $$ by $$ x-2 $$ using synthetic division, list the coefficients:
   $$
   1,\quad -3,\quad -6,\quad 0,\quad 6.
   $$

3. Set up synthetic division with $$ 2 $$ (since $$ x - 2 = 0 $$ when $$ x=2 $$):

- Bring down the 1.
   - Multiply $$ 1 $$ by $$ 2 $$ to get $$ 2 $$. Add to $$-3$$ to obtain $$-1$$.
   - Multiply $$-1$$ by $$ 2 $$ to get $$-2$$. Add to $$-6$$ to obtain $$-8$$.
   - Multiply $$-8$$ by $$ 2 $$ to get $$-16$$. Add to $$ 0 $$ to obtain $$-16$$.
   - Multiply $$-16$$ by $$ 2 $$ to get $$-32$$. Add to $$ 6 $$ to obtain $$-26$$.

4. The synthetic division yields:
   - **Quotient:** The coefficients $$ 1, -1, -8, -16 $$ correspond to the polynomial
     $$
     Q(x)=x^3 - x^2 - 8x - 16.
     $$
   - **Remainder:** $$ -26 $$.

5. By the Remainder Theorem, $$ P(2) $$ is equal to the remainder. Thus,
   $$
   P(2) = -26.
   $$

Quotient $$ = x^3 - x^2 - 8x - 16 $$  
Remainder $$ = -26 $$
Quotient $$ = x^3 - x^2 - 8x - 16 $$ Remainder $$ = -26 $$

Use the remainder theorem to find for .
Specifically, give the quotient and the remainder for the associated division and the value of .
Quotient
Remainder

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Use the remainder theorem to find for . Specifically, give the quotient and the remainder for the associated division and the value of . Quotient Remainder