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Use the long division method to find the result when 8, x, cubed, plus, 28, x, squared, plus, 14, x, minus, 138x 
3
 +28x 
2
 +14x−13 is divided by 2, x, plus, 32x+3. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ 
b(x)
r(x)
​
 .

We are tasked with dividing \(8x^3 + 28x^2 + 14x - 13\) by \(2x + 3\) using the long division method. Let's work through this step by step.

### Step 1: Set up the division
We want to divide the polynomial \(8x^3 + 28x^2 + 14x - 13\) (the **dividend**) by \(2x + 3\) (the **divisor**).

### Step 2: Divide the leading terms
The leading term of the dividend is \(8x^3\), and the leading term of the divisor is \(2x\). We begin by dividing these leading terms:

\[
\frac{8x^3}{2x} = 4x^2
\]

This tells us that the first term of the quotient is \(4x^2\).

### Step 3: Multiply and subtract
Next, we multiply \(4x^2\) by the entire divisor \(2x + 3\):

\[
4x^2 \times (2x + 3) = 8x^3 + 12x^2
\]

Now subtract this from the original polynomial:

\[
(8x^3 + 28x^2 + 14x - 13) - (8x^3 + 12x^2) = 16x^2 + 14x - 13
\]

### Step 4: Divide the leading terms again
Now, divide the leading term of the new polynomial \(16x^2\) by the leading term of the divisor \(2x\):

\[
\frac{16x^2}{2x} = 8x
\]

So the next term in the quotient is \(8x\).

### Step 5: Multiply and subtract again
Multiply \(8x\) by the divisor \(2x + 3\):

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

Subtract this from the current polynomial:

\[
(16x^2 + 14x - 13) - (16x^2 + 24x) = -10x - 13
\]

### Step 6: Divide the leading terms once more
Now, divide the leading term of the new polynomial \(-10x\) by the leading term of the divisor \(2x\):

\[
\frac{-10x}{2x} = -5
\]

So the next term in the quotient is \(-5\).

### Step 7: Multiply and subtract one last time
Multiply \(-5\) by the divisor \(2x + 3\):

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

Subtract this from the current polynomial:

\[
(-10x - 13) - (-10x - 15) = 2
\]

### Step 8: Final result
At this point, we have a remainder of 2. Thus, the division process is complete, and we can express the result in the form:

\[
q(x) = 4x^2 + 8x - 5, \quad r(x) = 2, \quad b(x) = 2x + 3
\]

So, the final answer is:

\[
\frac{8x^3 + 28x^2 + 14x - 13}{2x + 3} = 4x^2 + 8x - 5 + \frac{2}{2x + 3}
\]

Would you like further details on any specific step, or do you have any questions?

---

Here are 5 related questions for further exploration:

1. How does synthetic division compare with long division for polynomials?
2. Can polynomial division yield a zero remainder? What does that imply about the divisor and dividend?
3. How would you handle division by a higher-degree polynomial?
4. What is the relationship between the quotient and remainder in polynomial division?
5. How can polynomial division be used in finding factors of a polynomial?

**Tip:** Always check your final remainder to ensure it has a degree lower than that of the divisor when performing polynomial division!

Use the long division method to find the result when 8x^3 + 28x^2 + 14x - 13 is divided by 2x + 3. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Learn how to perform polynomial division using long division to divide 8x^3 + 28x^2 + 14x - 13 by 2x + 3. This step-by-step guide covers key algebra concepts, including quotient and remainder expressions. Suitable for Grades 10-12 students.

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