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

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 \(x - 6\).