The polynomial 𝑝(𝑥)=3𝑥3+𝑥2+𝑎𝑥+𝑏
 has (𝑥−2)
 as a factor.

When 𝑝(𝑥)
 is divided by 𝑥+2
, the remainder is 32
.

Find 𝑎
 and 𝑏
.

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.

The polynomial p(x) = 3x^3 + x^2 + ax + b has (x - 2) as a factor. When p(x) is divided by x + 2, the remainder is 32. Find a and b.

Learn how to solve a polynomial problem using the Factor and Remainder Theorems. Given p(x) = 3x^3 + x^2 + ax + b with (x - 2) as a factor and a remainder of 32 when divided by x + 2, find the values of a and b. Step-by-step solution included. Suitable for high school students in Grades 10-12.

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