Divide. $$ \frac{2 x^{4}-16 x^{3}+25 x^{2}-7 x+6}{x-6} $$

Question

UpStudy AI · Accepted Answer

To divide the polynomial $$ \frac{2x^{4} - 16x^{3} + 25x^{2} - 7x + 6}{x - 6} $$, we can use **synthetic division**. Here's a step-by-step breakdown:

### Step 1: Set Up Synthetic Division
We are dividing by $$ x - 6 $$, so we use $$ x = 6 $$ for the synthetic division.

**Coefficients of the dividend polynomial:**
$$ 2 \quad -16 \quad 25 \quad -7 \quad 6 $$

### Step 2: Perform Synthetic Division

1. **Bring down the first coefficient:**
   $$ 2 $$

2. **Multiply by 6 and add to the next coefficient:**
   $$ 2 \times 6 = 12 $$
   $$ -16 + 12 = -4 $$

3. **Multiply by 6 and add to the next coefficient:**
   $$ -4 \times 6 = -24 $$
   $$ 25 + (-24) = 1 $$

4. **Multiply by 6 and add to the next coefficient:**
   $$ 1 \times 6 = 6 $$
   $$ -7 + 6 = -1 $$

5. **Multiply by 6 and add to the last coefficient:**
   $$ -1 \times 6 = -6 $$
   $$ 6 + (-6) = 0 $$

### Step 3: Write the Result

The synthetic division yields:
$$ 2x^{3} - 4x^{2} + x - 1 $$
with a **remainder** of **0**.

### Conclusion

The division simplifies to:
$$ \frac{2x^{4} - 16x^{3} + 25x^{2} - 7x + 6}{x - 6} = 2x^{3} - 4x^{2} + x - 1 $$

**Final Answer:**
$$ 2x^{3} - 4x^{2} + x - 1 $$
The division simplifies to $$ 2x^{3} - 4x^{2} + x - 1 $$.

Divide.