Question

Which of the following expressions is not a factor of 3x3−10x2−143x−90 ? Use long division to solve.(1 point) Responses 3x+2 3 x plus 2 x−9 x minus 9 x−6 x minus 6 x+5

Answer 1

To determine which expression is not a factor of the polynomial \(3x^3 - 10x^2 - 143x - 90\), we can use polynomial long division for each expression option you provided: \(3x + 2\), \(x - 9\), \(x - 6\), and \(x + 5\).

Let’s start with polynomial long division for each expression:

1. **Dividing by \(3x + 2\):**
- Divide \(3x^3\) by \(3x\) to get \(x^2\).
- Multiply \(x^2\) by \(3x + 2\) to get \(3x^3 + 2x^2\).
- Subtract \(3x^3 + 2x^2\) from \(3x^3 - 10x^2 - 143x - 90\) to get \(-12x^2 - 143x - 90\).
- Divide \(-12x^2\) by \(3x\) to get \(-4x\).
- Multiply \(-4x\) by \(3x + 2\) to get \(-12x^2 - 8x\).
- Subtract \(-12x^2 - 8x\) from \(-12x^2 - 143x - 90\) to get \(-135x - 90\).
- Divide \(-135x\) by \(3x\) to get \(-45\).
- Multiply \(-45\) by \(3x + 2\) to get \(-135x - 90\).
- Subtract to find the remainder: \(0\).
- Result: \(3x + 2\) is a factor.

2. **Dividing by \(x - 9\):**
- Divide \(3x^3\) by \(x\) to get \(3x^2\).
- Multiply \(3x^2\) by \(x - 9\) to get \(3x^3 - 27x^2\).
- Subtract to get \(17x^2 - 143x - 90\).
- Divide \(17x^2\) by \(x\) to get \(17x\).
- Multiply \(17x\) by \(x - 9\) to get \(17x^2 - 153x\).
- Subtract to get \(10x - 90\).
- Divide \(10x\) by \(x\) to get \(10\).
- Multiply \(10\) by \(x - 9\) to get \(10x - 90\).
- Subtract to find the remainder: \(0\).
- Result: \(x - 9\) is a factor.

3. **Dividing by \(x - 6\):**
- Divide \(3x^3\) by \(x\) to get \(3x^2\).
- Multiply \(3x^2\) by \(x - 6\) to get \(3x^3 - 18x^2\).
- Subtract to get \(8x^2 - 143x - 90\).
- Divide \(8x^2\) by \(x\) to get \(8x\).
- Multiply \(8x\) by \(x - 6\) to get \(8x^2 - 48x\).
- Subtract to get \(-95x - 90\).
- Divide \(-95x\) by \(x\) to get \(-95\).
- Multiply \(-95\) by \(x - 6\) to get \(-95x + 570\).
- Subtract to find the remainder: \(-660\) (not zero).
- Result: \(x - 6\) is **not** a factor.

4. **Dividing by \(x + 5\):**
- Divide \(3x^3\) by \(x\) to get \(3x^2\).
- Multiply \(3x^2\) by \(x + 5\) to get \(3x^3 + 15x^2\).
- Subtract to get \(-25x^2 - 143x - 90\).
- Divide \(-25x^2\) by \(x\) to get \(-25x\).
- Multiply \(-25x\) by \(x + 5\) to get \(-25x^2 - 125x\).
- Subtract to get \(-18x - 90\).
- Divide \(-18x\) by \(x\) to get \(-18\).
- Multiply \(-18\) by \(x + 5\) to get \(-18x - 90\).
- Subtract to find the remainder: \(0\).
- Result: \(x + 5\) is a factor.

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