Two polynomials P and D are given. Use either synthetic division or long division to divide P(x) by D(x), and express the quotient
P(x)⁄D(x)
in the form
P(x)
D(x)
= Q(x) +
R(x)
D(x)
.
P(x) = 15x3 − 14x2 − x − 1      D(x) = 3x + 2

We are given the polynomials:

- \( P(x) = 15x^3 - 14x^2 - x - 1 \)
- \( D(x) = 3x + 2 \)

We need to divide \( P(x) \) by \( D(x) \) using either synthetic division or long division and express the result in the form:

\[
\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}
\]

### Step 1: Perform Polynomial Long Division

1. **Divide the leading term of \( P(x) \) by the leading term of \( D(x) \)**:  
   \[
   \frac{15x^3}{3x} = 5x^2
   \]
   So, the first term in the quotient \( Q(x) \) is \( 5x^2 \).

2. **Multiply \( 5x^2 \) by \( D(x) = 3x + 2 \)**:  
   \[
   5x^2(3x + 2) = 15x^3 + 10x^2
   \]
   Subtract this from \( P(x) \):
   \[
   (15x^3 - 14x^2 - x - 1) - (15x^3 + 10x^2) = -24x^2 - x - 1
   \]

3. **Divide the leading term of the new polynomial by the leading term of \( D(x) \)**:  
   \[
   \frac{-24x^2}{3x} = -8x
   \]
   So, the next term in the quotient \( Q(x) \) is \( -8x \).

4. **Multiply \( -8x \) by \( D(x) = 3x + 2 \)**:  
   \[
   -8x(3x + 2) = -24x^2 - 16x
   \]
   Subtract this from the current polynomial:
   \[
   (-24x^2 - x - 1) - (-24x^2 - 16x) = 15x - 1
   \]

5. **Divide the leading term of the new polynomial by the leading term of \( D(x) \)**:  
   \[
   \frac{15x}{3x} = 5
   \]
   So, the next term in the quotient \( Q(x) \) is \( 5 \).

6. **Multiply \( 5 \) by \( D(x) = 3x + 2 \)**:  
   \[
   5(3x + 2) = 15x + 10
   \]
   Subtract this from the current polynomial:
   \[
   (15x - 1) - (15x + 10) = -11
   \]

### Step 2: Write the Final Result

Now that we've completed the division, we can write the result in the form:

\[
\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}
\]

Where:
- \( Q(x) = 5x^2 - 8x + 5 \)
- \( R(x) = -11 \)

Thus, the quotient is:

\[
\frac{P(x)}{D(x)} = 5x^2 - 8x + 5 + \frac{-11}{3x + 2}
\]

Would you like more details or have any questions?

---

Here are 5 related questions to extend your understanding:

1. How is synthetic division different from long division when dividing polynomials?
2. Can you apply this division method to any polynomials of any degree?
3. What happens when the remainder \( R(x) \) is zero?
4. How can you use the quotient and remainder to verify your answer?
5. What is the significance of the remainder in polynomial division?

**Tip:** When dividing polynomials, always check that the degrees of both the divisor and the dividend are decreasing in power, which ensures correct alignment for the division process.

Two polynomials P and D are given. Use either synthetic division or long division to divide P(x) by D(x), and express the quotient P(x)⁄D(x) in the form P(x)/D(x) = Q(x) + R(x)/D(x). P(x) = 15x^3 − 14x^2 − x − 1 D(x) = 3x + 2

Learn how to divide the polynomial 15x^3 − 14x^2 − x − 1 by 3x + 2 using long division. This step-by-step solution explains the division process and how to express the quotient and remainder. Suitable for Grades 10-12 students studying polynomial division.

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