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

To divide the polynomial \( P(x) = 15x^3 - 14x^2 - x - 1 \) by \( D(x) = 3x + 2 \), we can use **synthetic division** or **long division**. I’ll proceed with long division, as it's a traditional and clear method for polynomial division.

### Step 1: Set up the division
We need to divide:

\[
\frac{15x^3 - 14x^2 - x - 1}{3x + 2}
\]

### Step 2: Perform the division
We divide the leading term of the numerator by the leading term of the denominator:

\[
\frac{15x^3}{3x} = 5x^2
\]

Now, multiply \( 5x^2 \) by \( 3x + 2 \):

\[
5x^2(3x + 2) = 15x^3 + 10x^2
\]

Subtract this from the original polynomial:

\[
(15x^3 - 14x^2) - (15x^3 + 10x^2) = -24x^2
\]

### Step 3: Continue the division
Now divide the new leading term by the leading term of the denominator:

\[
\frac{-24x^2}{3x} = -8x
\]

Multiply \( -8x \) by \( 3x + 2 \):

\[
-8x(3x + 2) = -24x^2 - 16x
\]

Subtract this from the current polynomial:

\[
(-24x^2 - x) - (-24x^2 - 16x) = 15x
\]

### Step 4: Continue the division
Now divide the new leading term by the leading term of the denominator:

\[
\frac{15x}{3x} = 5
\]

Multiply \( 5 \) by \( 3x + 2 \):

\[
5(3x + 2) = 15x + 10
\]

Subtract this from the current polynomial:

\[
(15x - 1) - (15x + 10) = -11
\]

### Step 5: Write the final result
We have completed the division. The quotient is \( Q(x) = 5x^2 - 8x + 5 \) and the remainder is \( R(x) = -11 \).

Thus, we can express the result in the form:

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

Substituting the values:

\[
\frac{15x^3 - 14x^2 - x - 1}{3x + 2} = 5x^2 - 8x + 5 + \frac{-11}{3x + 2}
\]

This is the final result.

---

Would you like more details on any part of the process or explanation?

Here are 5 relative questions you can explore:
1. How is synthetic division different from long division in polynomials?
2. Can synthetic division be used when the divisor is not a linear polynomial?
3. How can we verify the result of a polynomial division?
4. What are the applications of polynomial division in calculus?
5. How do we factorize higher-degree polynomials using division?

**Tip:** Always check your remainder when dividing polynomials—it can help verify if the solution is correct.

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 P(x) = 15x^3 − 14x^2 − x − 1 by D(x) = 3x + 2 using long division. This step-by-step solution explains key concepts like polynomial division, remainders, and how to express the quotient in the form Q(x) + R(x)/D(x). Suitable for Grades 10-12.

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