Question

Which synthetic division shows how to find the quotient?

8x3+6x2−18x+4÷4x−1


(1 point)

Answer 1

To perform synthetic division for the polynomial division \( 8x^3 + 6x^2 - 18x + 4 \) by \( 4x - 1 \), we first need to convert the divisor \( 4x - 1 \) into the form \( x - r \). We find \( r \) by setting \( 4x - 1 = 0 \):

\[
4x - 1 = 0 \implies 4x = 1 \implies x = \frac{1}{4}
\]

Now we can use synthetic division with \( r = \frac{1}{4} \). Here are the steps to perform synthetic division:

1. Write down the coefficients of the dividend. The polynomial \( 8x^3 + 6x^2 - 18x + 4 \) has the coefficients \( 8, 6, -18, 4 \).

2. Set up synthetic division as follows, placing \( \frac{1}{4} \) on the left and the coefficients on the right:

```
1/4 | 8 6 -18 4
|
```

3. Bring down the leading coefficient:

```
1/4 | 8 6 -18 4
|
|___________________
8
```

4. Multiply \( 8 \) by \( \frac{1}{4} \) and write the result under the next coefficient:

```
1/4 | 8 6 -18 4
| 2
|___________________
8 8
```

5. Add the second column:

```
1/4 | 8 6 -18 4
| 2
|___________________
8 8
```

6. Continue this process. Multiply \( 8 \) by \( \frac{1}{4} \) and add to the next coefficient:

```
1/4 | 8 6 -18 4
| 2 2
|___________________
8 8 -16
```

7. Multiply \( -16 \) by \( \frac{1}{4} \) and add to \( 4 \):

```
1/4 | 8 6 -18 4
| 2 2 -4
|___________________
8 8 -16 0
```

8. The result shows the coefficients of the quotient polynomial and the remainder. The quotient is \( 8x^2 + 8x - 16 \) and the remainder is \( 0 \).

So, the result of the division \( 8x^3 + 6x^2 - 18x + 4 \) divided by \( 4x - 1 \) is:

\[
8x^2 + 8x - 16
\]

with no remainder.