Question

Which of the following expressions is not a factor of 3x3−10x2−143x−90 ? Use long division to solve.(1 point)
Responses

x+5
x plus 5

3x+2
3 x plus 2

x−6
x minus 6

x−9

Answer 1

To determine which of the expressions is not a factor of the polynomial \(3x^3 - 10x^2 - 143x - 90\), we can use polynomial long division.

Let's check each option one by one.

### 1. Testing \(x + 5\)

To perform long division, we divide \(3x^3 - 10x^2 - 143x - 90\) by \(x + 5\).

1. **Divide**: \(3x^3 \div x = 3x^2\).
2. **Multiply**: \(3x^2(x + 5) = 3x^3 + 15x^2\).
3. **Subtract**: \((3x^3 - 10x^2) - (3x^3 + 15x^2) = -25x^2 - 143x\).

4. **Bring down**: \(-25x^2 - 143x\).
5. **Divide**: \(-25x^2 \div x = -25x\).
6. **Multiply**: \(-25x(x + 5) = -25x^2 - 125x\).
7. **Subtract**: \((-25x^2 - 143x) - (-25x^2 - 125x) = -18x\).

8. **Bring down**: \(-18x - 90\).
9. **Divide**: \(-18x \div x = -18\).
10. **Multiply**: \(-18(x + 5) = -18x - 90\).
11. **Subtract**: \((-18x - 90) - (-18x - 90) = 0\).

Since the remainder is \(0\), \(x + 5\) is a factor.

### 2. Testing \(3x + 2\)

Next, we test \(3x + 2\).

1. **Divide**: \(3x^3 \div 3x = x^2\).
2. **Multiply**: \(x^2(3x + 2) = 3x^3 + 2x^2\).
3. **Subtract**: \((3x^3 - 10x^2) - (3x^3 + 2x^2) = -12x^2 - 143x\).

4. **Bring down**: \(-12x^2 - 143x\).
5. **Divide**: \(-12x^2 \div 3x = -4x\).
6. **Multiply**: \(-4x(3x + 2) = -12x^2 - 8x\).
7. **Subtract**: \((-12x^2 - 143x) - (-12x^2 - 8x) = -135x\).

8. **Bring down**: \(-135x - 90\).
9. **Divide**: \(-135x \div 3x = -45\).
10. **Multiply**: \(-45(3x + 2) = -135x - 90\).
11. **Subtract**: \((-135x - 90) - (-135x - 90) = 0\).

Since the remainder is \(0\), \(3x + 2\) is a factor.

### 3. Testing \(x - 6\)

Now we test \(x - 6\).

1. **Divide**: \(3x^3 \div x = 3x^2\).
2. **Multiply**: \(3x^2(x - 6) = 3x^3 - 18x^2\).
3. **Subtract**: \((3x^3 - 10x^2) - (3x^3 - 18x^2) = 8x^2 - 143x\).

4. **Bring down**: \(8x^2 - 143x\).
5. **Divide**: \(8x^2 \div x = 8x\).
6. **Multiply**: \(8x(x - 6) = 8x^2 - 48x\).
7. **Subtract**: \((8x^2 - 143x) - (8x^2 - 48x) = -95x\).

8. **Bring down**: \(-95x - 90\).
9. **Divide**: \(-95x \div x = -95\).
10. **Multiply**: \(-95(x - 6) = -95x + 570\).
11. **Subtract**: \((-95x - 90) - (-95x + 570) = -660\).

The remainder is \(-660\), so \(x - 6\) is **not** a factor.

### 4. Testing \(x - 9\)

Finally, let's test \(x - 9\).

1. **Divide**: \(3x^3 \div x = 3x^2\).
2. **Multiply**: \(3x^2(x - 9) = 3x^3 - 27x^2\).
3. **Subtract**: \((3x^3 - 10x^2) - (3x^3 - 27x^2) = 17x^2 - 143x\).

4. **Bring down**: \(17x^2 - 143x\).
5. **Divide**: \(17x^2 \div x = 17x\).
6. **Multiply**: \(17x(x - 9) = 17x^2 - 153x\).
7. **Subtract**: \((17x^2 - 143x) - (17x^2 - 153x) = 10x\).

8. **Bring down**: \(10x - 90\).
9. **Divide**: \(10x \div x = 10\).
10. **Multiply**: \(10(x - 9) = 10x - 90\).
11. **Subtract**: \((10x - 90) - (10x - 90) = 0\).

Since the remainder is \(0\), \(x - 9\) is also a factor.

### Conclusion

The option that is **not** a factor of \(3x^3 - 10x^2 - 143x - 90\) is \(x - 6\).