Question

Using long division, determine which of the following polynomials divides evenly for a divisor of x+3
𝑥
+
3
.(1 point)
Responses

x3−5x2+10x−15
𝑥
3
−
5
𝑥
2
+
10
𝑥
−
15
x cubed minus 5 x squared plus 10 x minus 15

3x2−6x+9
3
𝑥
2
−
6
𝑥
+
9
3 x squared minus 6 x plus 9

5x2+7x−12
5
𝑥
2
+
7
𝑥
−
12
5 x squared plus 7 x minus 12

x3−3x2−13x+15
𝑥
3
−
3
𝑥
2
−
13
𝑥
+
15
x cubed minus 3 x squared minus 13 x plus 15

Answer 1

To determine which of the given polynomials divides evenly by \( x + 3 \), we can use synthetic division. If the remainder is zero, then the divisor divides the polynomial evenly.

In synthetic division, we use the root of the divisor \( x + 3 = 0 \) or \( x = -3 \) to perform the division.

Let's evaluate each polynomial one by one using synthetic division.

### 1. Polynomial: \( x^3 - 5x^2 + 10x - 15 \)

Using \( -3 \):

```
-3 | 1 -5 10 -15
| -3 24 -90
------------------------
1 -8 34 -105
```

Remainder: \( -105 \) (not divisible)

### 2. Polynomial: \( 3x^2 - 6x + 9 \)

Using \( -3 \):

```
-3 | 3 -6 9
| -9 45
---------------
3 -15 54
```

Remainder: \( 54 \) (not divisible)

### 3. Polynomial: \( 5x^2 + 7x - 12 \)

Using \( -3 \):

```
-3 | 5 7 -12
| -15 24
---------------
5 -8 12
```

Remainder: \( 12 \) (not divisible)

### 4. Polynomial: \( x^3 - 3x^2 - 13x + 15 \)

Using \( -3 \):

```
-3 | 1 -3 -13 15
| -3 18 -45
---------------------
1 -6 5 -30
```

Remainder: \( -30 \) (not divisible)

### Conclusion

None of the given polynomials divide evenly by \( x + 3 \). All of them yield a non-zero remainder.