Use the long division method to find the result when 2, x, cubed, plus, 21, x, squared, plus, 2, x, minus, 42x 
3
 +21x 
2
 +2x−4 is divided by 2, x, plus, 12x+1. 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 asked to divide \( 2x^3 + 21x^2 + 2x - 4 \) by \( 2x + 1 \) using the **long division method**.

### Step 1: Setup the division
We are dividing:

\[
\frac{2x^3 + 21x^2 + 2x - 4}{2x + 1}
\]

The dividend is \( 2x^3 + 21x^2 + 2x - 4 \), and the divisor is \( 2x + 1 \).

### Step 2: Divide the leading terms
Divide the leading term of the dividend by the leading term of the divisor:

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

So, the first term of the quotient is \( x^2 \).

### Step 3: Multiply and subtract
Now, multiply \( x^2 \) by the divisor \( 2x + 1 \):

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

Subtract this from the original polynomial:

\[
(2x^3 + 21x^2 + 2x - 4) - (2x^3 + x^2) = (21x^2 - x^2) + 2x - 4 = 20x^2 + 2x - 4
\]

### Step 4: Divide the next leading term
Now divide the leading term \( 20x^2 \) by \( 2x \):

\[
\frac{20x^2}{2x} = 10x
\]

So, the next term of the quotient is \( 10x \).

### Step 5: Multiply and subtract
Multiply \( 10x \) by the divisor \( 2x + 1 \):

\[
10x \cdot (2x + 1) = 20x^2 + 10x
\]

Subtract this from \( 20x^2 + 2x - 4 \):

\[
(20x^2 + 2x - 4) - (20x^2 + 10x) = (2x - 10x) - 4 = -8x - 4
\]

### Step 6: Divide the next leading term
Now divide \( -8x \) by \( 2x \):

\[
\frac{-8x}{2x} = -4
\]

So, the next term of the quotient is \( -4 \).

### Step 7: Multiply and subtract
Multiply \( -4 \) by the divisor \( 2x + 1 \):

\[
-4 \cdot (2x + 1) = -8x - 4
\]

Subtract this from \( -8x - 4 \):

\[
(-8x - 4) - (-8x - 4) = 0
\]

### Step 8: Final answer
The remainder is zero, meaning the division is exact. So the quotient is:

\[
x^2 + 10x - 4
\]

Thus, the result of the division is:

\[
\frac{2x^3 + 21x^2 + 2x - 4}{2x + 1} = x^2 + 10x - 4
\]

There is no remainder, so the result is simply \( x^2 + 10x - 4 \).

---

Do you need further details or have any questions?

Here are 5 related questions:
1. How do we handle long division if there is a remainder?
2. Can we divide polynomials with terms missing in the middle, like \( 2x^3 - 4 \)?
3. How do synthetic division and long division compare for polynomials?
4. What happens if we divide by a polynomial with a degree higher than 1?
5. Can polynomial division be used to factor higher-degree polynomials?

**Tip:** Always align terms by their degrees when performing long division to avoid confusion.

Use the long division method to find the result when 2x^3 + 21x^2 + 2x - 4 is divided by 2x + 1. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Learn how to divide 2x^3 + 21x^2 + 2x - 4 by 2x + 1 using the long division method. This detailed step-by-step solution explores polynomial division concepts, the division algorithm, and algebraic techniques. Suitable for students in Grades 10-12.

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