Question

Divide using synthetic division.

(2x^3+3x^2-17x-30)/(x-2)




A
2x2+7x−3+36x−22x^2+7x-3+\frac{36}{x-2}2x
2
+7x−3+
x−2
36
​


B
2x2−x−152x^2-x-152x
2
−x−15

C
2x2+7x−3−36x−22x^2+7x-3-\frac{36}{x-2}2x
2
+7x−3−
x−2
36
​


D
2x2+7x+3−24x−22x^2+7x+3-\frac{24}{x-2}2x
2
+7x+3−
x−2
24
​

Answer 1

To divide \(2x^3 + 3x^2 - 17x - 30\) by \(x - 2\) using synthetic division, we will follow these steps:

1. Set up synthetic division with the root of \(x - 2\), which is \(2\).
2. Write down the coefficients of the polynomial: \(2, 3, -17, -30\).
3. Carry down the leading coefficient and perform the synthetic division process.

### Step 1: Set up synthetic division
The coefficients are:
- 2 (for \(2x^3\))
- 3 (for \(3x^2\))
- -17 (for \(-17x\))
- -30 (for the constant term)

### Step 2: Synthetic division calculation
```
2 | 2 3 -17 -30
| 4 14 -6
---------------------
2 7 -3 -36
```

1. Bring down the \(2\).
2. Multiply \(2\) (the number we are dividing by) by \(2\) (which we just brought down): \(2 \times 2 = 4\), and add it to \(3\): \(3 + 4 = 7\).
3. Multiply \(2\) by \(7\): \(2 \times 7 = 14\), and add it to \(-17\): \(-17 + 14 = -3\).
4. Multiply \(2\) by \(-3\): \(2 \times -3 = -6\), and add it to \(-30\): \(-30 - 6 = -36\).

### Step 3: Write the result
The synthetic division has provided us with a quotient of \(2x^2 + 7x - 3\) and a remainder of \(-36\). So, we can express the result as:

\[
2x^2 + 7x - 3 - \frac{36}{x - 2}
\]

Now we can compare this result with the choices given:

- **A:** \(2x^2 + 7x - 3 + \frac{36}{x - 2}\)
- **B:** \(2x^2 - x - 15\)
- **C:** \(2x^2 + 7x - 3 - \frac{36}{x - 2}\) (This matches our result!)
- **D:** \(2x^2 + 7x + 3 - \frac{24}{x - 2}\)

The correct choice is **C**:
\[
\boxed{C}
\]
which represents the result of the synthetic division accurately.