Question

Use synthetic division to divide the polynomial 3x4+2x2−5x+6
by x−1
.

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Answer 1

To use synthetic division to divide the polynomial \( 3x^4 + 0x^3 + 2x^2 - 5x + 6 \) by \( x - 1 \), we first note the root of the divisor \( x - 1 \), which is \( 1 \).

### Step 1: Write down the coefficients
Write down the coefficients of the polynomial:

- Coefficient of \( x^4 \): 3
- Coefficient of \( x^3 \): 0 (since there is no \( x^3 \) term)
- Coefficient of \( x^2 \): 2
- Coefficient of \( x^1 \): -5
- Constant term: 6

Thus, the coefficients are:
\[ 3, 0, 2, -5, 6 \]

### Step 2: Set up Synthetic Division
We set up the synthetic division using the root \( 1 \).

```
1 | 3 0 2 -5 6
|
```

### Step 3: Perform the synthetic division
1. Bring down the first coefficient (3):

```
1 | 3 0 2 -5 6
|
| 3
```

2. Multiply the value you just wrote down by the divisor root (1):
- \( 3 \times 1 = 3 \)

3. Add this value to the next coefficient (0):
- \( 0 + 3 = 3 \)

```
1 | 3 0 2 -5 6
| 3
| 3 3
```

4. Repeat steps 2 and 3 for the remaining coefficients:
- Multiply \( 3 \times 1 = 3 \), then add to 2:
- \( 2 + 3 = 5 \)
- Multiply \( 5 \times 1 = 5 \), then add to -5:
- \( -5 + 5 = 0 \)
- Multiply \( 0 \times 1 = 0 \), then add to 6:
- \( 6 + 0 = 6 \)

```
1 | 3 0 2 -5 6
| 3 5 0
| 3 3 5 0 6
```

### Step 4: Write down the result
The bottom row contains the coefficients of the quotient polynomial and the remainder.

The quotient polynomial is:
- Coefficients: \( 3, 3, 5, 0 \) corresponds to \( 3x^3 + 3x^2 + 5x + 0\)

The remainder is \( 6 \).

### Final result
Thus, we can express the original division as:
\[
3x^4 + 2x^2 - 5x + 6 = (x - 1)(3x^3 + 3x^2 + 5x) + 6
\]

**Quotient:** \( 3x^3 + 3x^2 + 5x \)
**Remainder:** \( 6 \)