Given the polynomial $p(x) = 3x^3 + x^2 + ax + b$, you know two key pieces of information:

1. $(x - 2)$ is a factor of $p(x)$. This means that $p(2) = 0$.
2. When $p(x)$ is divided by $x + 2$, the remainder is 32. This means that $p(-2) = 32$.

Using these, we can set up two equations to solve for $a$ and $b$.

Step 1: Use the condition $p(2) = 0$

Substitute $x = 2$ into the polynomial:

$p(2) = 3(2)^3 + (2)^2 + a(2) + b = 0$ $3(8) + 4 + 2a + b = 0$ $24 + 4 + 2a + b = 0$ $28 + 2a + b = 0$ $2a + b = -28 \quad \text{(Equation 1)}$

Step 2: Use the condition $p(-2) = 32$

Substitute $x = -2$ into the polynomial:

$p(-2) = 3(-2)^3 + (-2)^2 + a(-2) + b = 32$ $3(-8) + 4 - 2a + b = 32$ $-24 + 4 - 2a + b = 32$ $-20 - 2a + b = 32$ $-2a + b = 52 \quad \text{(Equation 2)}$

Step 3: Solve the system of equations

Now we have the system:

1. $2a + b = -28$
2. $-2a + b = 52$

To solve this, subtract Equation 1 from Equation 2:

$(-2a + b) - (2a + b) = 52 - (-28)$ $-4a = 80$ $a = -20$

Step 4: Substitute $a = -20$ into Equation 1

Substitute $a = -20$ into $2a + b = -28$:

$2(-20) + b = -28$ $-40 + b = -28$ $b = 12$

Final Answer:

• $a = -20$
• $b = 12$

Would you like a detailed explanation on how to check this solution or ask any clarifying questions?

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Here are five questions related to the solution:

1. How do we verify if $(x - 2)$ is a factor of $p(x)$?
2. What is the Remainder Theorem and how is it applied in polynomial division?
3. How would the result change if $p(x)$ had a different degree?
4. What are alternative methods to solve this system of equations?
5. What are other conditions that can be given to determine the coefficients in a polynomial?

Tip: Always check your solution by substituting the values back into the original conditions to ensure accuracy.