Use the long division method to find the result when 12, x, cubed, plus, 10, x, squared, plus, 11, x, plus, 312x
3
+10x
2
+11x+3 is divided by 3, x, plus, 13x+1.

To solve the polynomial division of \( 12x^3 + 10x^2 + 11x + 3 \) by \( 3x + 1 \) using the long division method, let's go step by step.

### Step 1: Set up the division
Divide the leading term of the dividend by the leading term of the divisor.

- Dividend: \( 12x^3 + 10x^2 + 11x + 3 \)
- Divisor: \( 3x + 1 \)

First, divide the leading term of the dividend, \( 12x^3 \), by the leading term of the divisor, \( 3x \).

\[
\frac{12x^3}{3x} = 4x^2
\]

### Step 2: Multiply the entire divisor by this result
Multiply \( 4x^2 \) by \( 3x + 1 \):

\[
4x^2 \cdot (3x + 1) = 12x^3 + 4x^2
\]

### Step 3: Subtract this result from the original polynomial
Now subtract \( 12x^3 + 4x^2 \) from the original polynomial:

\[
(12x^3 + 10x^2 + 11x + 3) - (12x^3 + 4x^2) = 6x^2 + 11x + 3
\]

### Step 4: Repeat the process
Next, divide the leading term of the new polynomial \( 6x^2 \) by the leading term of the divisor \( 3x \):

\[
\frac{6x^2}{3x} = 2x
\]

Multiply \( 2x \) by the divisor \( 3x + 1 \):

\[
2x \cdot (3x + 1) = 6x^2 + 2x
\]

Subtract this from the current polynomial:

\[
(6x^2 + 11x + 3) - (6x^2 + 2x) = 9x + 3
\]

### Step 5: Continue the process
Now divide \( 9x \) by \( 3x \):

\[
\frac{9x}{3x} = 3
\]

Multiply \( 3 \) by \( 3x + 1 \):

\[
3 \cdot (3x + 1) = 9x + 3
\]

Subtract this from the current polynomial:

\[
(9x + 3) - (9x + 3) = 0
\]

### Final Result
The remainder is \( 0 \), and the quotient is \( 4x^2 + 2x + 3 \).

So, the result of the division is:

\[
\boxed{4x^2 + 2x + 3}
\]

Would you like more details on any part of this process, or do you have any questions?

Here are some related questions you might find helpful:
1. How do you handle cases where there's a remainder in polynomial division?
2. What is synthetic division and how does it compare to long division for polynomials?
3. How do you factor polynomials using the quotient obtained from division?
4. How is the remainder theorem applied in polynomial division?
5. Can polynomial division be applied to non-linear divisors?

**Tip:** Always align terms by their degrees in polynomial long division to avoid confusion.

Explore the step-by-step solution for polynomial division of 12x^3 + 10x^2 + 11x + 3 by 3x + 13x + 1 using the long division method. Learn the process of dividing polynomials, handling remainders, and finding the quotient. Suitable for students in Grades 10-12.

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