\begin{tabular}{|l} I can divide polynomials using the box method. \ Divide using the long division (box method). MUST SHOW WORK FOR FULL CREDIT. \ $$ \left(4 x^{3}+x-2 ight) \div(2 x-1) $$ \ \hline\end{tabular}

Question

\begin{tabular}{|l} I can divide polynomials using the box method. \ Divide using the long division (box method). MUST SHOW WORK FOR FULL CREDIT. \ $$ \left(4 x^{3}+x-2ight) \div(2 x-1) $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

To divide the polynomial $$4x^3 + x - 2$$ by $$2x - 1$$ using long division (box method), we will follow these steps:

### Step 1: Set Up the Division
We will set up the long division as follows:

```
          ____________
2x - 1 | 4x^3 + 0x^2 + x - 2
```

### Step 2: Divide the Leading Terms
1. Divide the leading term of the dividend $$4x^3$$ by the leading term of the divisor $$2x$$:
   $$
   \frac{4x^3}{2x} = 2x^2
   $$

### Step 3: Multiply and Subtract
2. Multiply $$2x^2$$ by the entire divisor $$2x - 1$$:
   $$
   2x^2(2x - 1) = 4x^3 - 2x^2
   $$
3. Subtract this result from the original polynomial:
   $$
   (4x^3 + 0x^2 + x - 2) - (4x^3 - 2x^2) = 0x^3 + 2x^2 + x - 2
   $$
   This simplifies to:
   $$
   2x^2 + x - 2
   $$

### Step 4: Repeat the Process
4. Now, divide the leading term $$2x^2$$ by $$2x$$:
   $$
   \frac{2x^2}{2x} = x
   $$
5. Multiply $$x$$ by the divisor $$2x - 1$$:
   $$
   x(2x - 1) = 2x^2 - x
   $$
6. Subtract this from the current polynomial:
   $$
   (2x^2 + x - 2) - (2x^2 - x) = 0x^2 + 2x - 2
   $$
   This simplifies to:
   $$
   2x - 2
   $$

### Step 5: Final Division
7. Divide the leading term $$2x$$ by $$2x$$:
   $$
   \frac{2x}{2x} = 1
   $$
8. Multiply $$1$$ by the divisor $$2x - 1$$:
   $$
   1(2x - 1) = 2x - 1
   $$
9. Subtract this from the current polynomial:
   $$
   (2x - 2) - (2x - 1) = 0x - 1
   $$
   This simplifies to:
   $$
   -1
   $$

### Step 6: Write the Final Result
The result of the division is:
$$
2x^2 + x + 1 - \frac{1}{2x - 1}
$$

### Final Answer
Thus, the final answer for the division $$ \left(4x^3 + x - 2\right) \div (2x - 1) $$ is:
$$
2x^2 + x + 1 - \frac{1}{2x - 1}
$$
The division of $$4x^3 + x - 2$$ by $$2x - 1$$ using the long division method results in: $$ 2x^2 + x + 1 - \frac{1}{2x - 1} $$

\begin{tabular}{|l} I can divide polynomials using the box method. \\ Divide using the long division (box method). MUST SHOW WORK FOR FULL CREDIT. \\ \( \left(4 x^{3}+x-2\right) \div(2 x-1) \) \\ \hline\end{tabular}

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