5. When the expression $$ x^{3}+p x^{2}+2 x+1 $$ is divided by $$ (x-2) $$, the remainder is thrice the remainder when the divisor is $$ (x+1) $$. Find the value of $$ p $$. [Answer Key] $$ p=-19 $$

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Given the polynomial 
$$
f(x) = x^3 + p x^2 + 2x + 1,
$$
we first find the remainders when the polynomial is divided by $$ (x-2) $$ and $$ (x+1) $$.

**Step 1.**  
When dividing by $$ (x-2) $$, the remainder is $$ f(2) $$:
$$
f(2) = 2^3 + p \cdot 2^2 + 2(2) + 1 = 8 + 4p + 4 + 1 = 4p + 13.
$$

**Step 2.**  
When dividing by $$ (x+1) $$, the remainder is $$ f(-1) $$:
$$
f(-1) = (-1)^3 + p \cdot (-1)^2 + 2(-1) + 1 = -1 + p - 2 + 1 = p - 2.
$$

**Step 3.**  
According to the problem, the remainder when divided by $$ (x-2) $$ is thrice the remainder when divided by $$ (x+1) $$. Thus, we have:
$$
4p + 13 = 3(p - 2).
$$

**Step 4.**  
Solve the equation step by step:
$$
4p + 13 = 3p - 6.
$$
Subtract $$ 3p $$ from both sides:
$$
p + 13 = -6.
$$
Subtract 13 from both sides:
$$
p = -19.
$$

So, the value of $$ p $$ is:
$$
\boxed{-19}.
$$
$$ p = -19 $$

When the expression is divided by , the remainder is thrice the
remainder when the divisor is . Find the value of .
[Answer Key]

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When the expression is divided by , the remainder is thrice the remainder when the divisor is . Find the value of . [Answer Key]