To determine which expression is not a factor of \(3x^3 - 10x^2 - 143x - 90\), we can use polynomial long division. We'll perform the division for each option and see which one does not divide evenly (produces a remainder).
Option A: \(x - 9\)
- Divide \(3x^3\) by \(x\) to get \(3x^2\).
- Multiply \(3x^2\) by \(x - 9\) to get \(3x^3 - 27x^2\).
- Subtract this from \(3x^3 - 10x^2 - 143x - 90\): \[ (3x^3 - 10x^2) - (3x^3 - 27x^2) = 17x^2. \]
- Bring down \(-143x - 90\) 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 this from \(17x^2 - 143x - 90\): \[ (-143x + 153x) = 10x - 90. \]
- Bring down \(-90\) to get \(10x - 90\).
- Divide \(10x\) by \(x\) to get \(10\).
- Multiply \(10\) by \(x - 9\) to get \(10x - 90\).
- Subtract to get a remainder of \(0\).
Option A \(x-9\) is a factor with no remainder.
Option B: \(x + 5\)
- Divide \(3x^3\) by \(x\) to get \(3x^2\).
- Multiply \(3x^2\) by \(x + 5\) to get \(3x^3 + 15x^2\).
- Subtract: \[ (3x^3 - 10x^2) - (3x^3 + 15x^2) = -25x^2. \]
- Bring down \(-143x - 90\) 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: \[ (-143x + 125x) = -18x - 90. \]
- Bring down \(-90\) to get \(-18x - 90\).
- Divide \(-18x\) by \(x\) to get \(-18\).
- Multiply \(-18\) by \(x + 5\) to get \(-18x - 90\).
- Remainder is \(0\) because the subtraction results in \(0\).
Option B \(x+5\) is a factor with no remainder.
Option C: \(x - 6\)
- Divide \(3x^3\) by \(x\) to get \(3x^2\).
- Multiply \(3x^2\) by \(x - 6\) to get \(3x^3 - 18x^2\).
- Subtract: \[ (-10x^2 + 18x^2) = 8x^2. \]
- Bring down \(-143x - 90\) 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: \[ (-143x + 48x) = -95x - 90. \]
- Bring down \(-90\) to get \(-95x - 90\).
- Divide \(-95x\) by \(x\) to get \(-95\).
- Multiply \(-95\) by \(x - 6\) to get \(-95x + 570\).
- Remainder: \[ -90 - 570 = -660. \]
Option C \(x-6\) has a remainder of \(-660\). It cannot be a factor.
Option D: \(3x + 2\)
- Divide \(3x^3\) by \(3x\) to get \(x^2\).
- Multiply \(x^2\) by \(3x + 2\) to get \(3x^3 + 2x^2\).
- Subtract: \[ (-10x^2 - 2x^2) = -12x^2. \]
- Bring down \(-143x - 90\) 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: \[ (-143x + 8x) = -135x - 90. \]
- Bring down \(-90\) to get \(-135x - 90\).
- Divide \(-135x\) by \(3x\) to get \(-45\).
- Multiply \(-45\) by \(3x + 2\) to get \(-135x - 90\).
Option D \(3x+2\) has no remainder.
Upon evaluating all the options using long division:
- A. \(x - 9\): Factor (No remainder)
- B. \(x + 5\): Factor (No remainder)
- C. \(x - 6\): Not a factor (Remainder \(-660\))
- D. \(3x + 2\): Factor (No remainder)
Conclusion:
The expression that is not a factor is C. \(x - 6\).
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