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Which of the following expressions is not a factor of 3x^3−10x^2−143x−90 ? Use long division to solve

a. X-9
b. X+5
c. 3x+2
d. X-6

1 answer

To determine which expression is not a factor of \(3x^3 - 10x^2 - 143x - 90\), we can use polynomial long division. If the division results in a remainder of zero, then the divisor is a factor of the polynomial.

Let's evaluate each option one by one using long division.

Option A: \(x - 9\)

  1. Divide \(3x^3\) by \(x\) to get \(3x^2\).
  2. Multiply \(3x^2\) by \(x - 9\) to get \(3x^3 - 27x^2\).
  3. Subtract: \[ (3x^3 - 10x^2 - 143x - 90) - (3x^3 - 27x^2) = 17x^2 - 143x - 90 \]
  4. Bring down the next term: \(17x^2 - 143x - 90\).
  5. Divide \(17x^2\) by \(x\) to get \(17x\).
  6. Multiply \(17x\) by \(x - 9\): \(17x^2 - 153x\).
  7. Subtract: \[ (17x^2 - 143x - 90) - (17x^2 - 153x) = 10x - 90 \]
  8. Divide \(10x\) by \(x\) to get \(10\).
  9. Multiply \(10\) by \(x - 9\): \(10x - 90\).
  10. Subtract: \[ (10x - 90) - (10x - 90) = 0 \]

Remainder is 0, so \(x - 9\) is a factor.

Option B: \(x + 5\)

  1. Divide \(3x^3\) by \(x\) to get \(3x^2\).
  2. Multiply \(3x^2\) by \(x + 5\): \(3x^3 + 15x^2\).
  3. Subtract: \[ (3x^3 - 10x^2 - 143x - 90) - (3x^3 + 15x^2) = -25x^2 - 143x - 90 \]
  4. Bring down the next term: \(-25x^2 - 143x - 90\).
  5. Divide \(-25x^2\) by \(x\) to get \(-25x\).
  6. Multiply \(-25x\) by \(x + 5\): \(-25x^2 - 125x\).
  7. Subtract: \[ (-25x^2 - 143x - 90) - (-25x^2 - 125x) = -18x - 90 \]
  8. Divide \(-18x\) by \(x\) to get \(-18\).
  9. Multiply \(-18\) by \(x + 5\): \(-18x - 90\).
  10. Subtract: \[ (-18x - 90) - (-18x - 90) = 0 \]

Remainder is 0, so \(x + 5\) is a factor.

Option C: \(3x + 2\)

  1. Divide \(3x^3\) by \(3x\) to get \(x^2\).
  2. Multiply \(x^2\) by \(3x + 2\): \(3x^3 + 2x^2\).
  3. Subtract: \[ (3x^3 - 10x^2 - 143x - 90) - (3x^3 + 2x^2) = -12x^2 - 143x - 90 \]
  4. Bring down the next term: \(-12x^2 - 143x - 90\).
  5. Divide \(-12x^2\) by \(3x\) to get \(-4x\).
  6. Multiply \(-4x\) by \(3x + 2\): \(-12x^2 - 8x\).
  7. Subtract: \[ (-12x^2 - 143x - 90) - (-12x^2 - 8x) = -135x - 90 \]
  8. Divide \(-135x\) by \(3x\) to get \(-45\).
  9. Multiply \(-45\) by \(3x + 2\): \(-135x - 90\).
  10. Subtract: \[ (-135x - 90) - (-135x - 90) = 0 \]

Remainder is 0, so \(3x + 2\) is a factor.

Option D: \(x - 6\)

  1. Divide \(3x^3\) by \(x\) to get \(3x^2\).
  2. Multiply \(3x^2\) by \(x - 6\): \(3x^3 - 18x^2\).
  3. Subtract: \[ (3x^3 - 10x^2 - 143x - 90) - (3x^3 - 18x^2) = 8x^2 - 143x - 90 \]
  4. Bring down the next term: \(8x^2 - 143x - 90\).
  5. Divide \(8x^2\) by \(x\) to get \(8x\).
  6. Multiply \(8x\) by \(x - 6\): \(8x^2 - 48x\).
  7. Subtract: \[ (8x^2 - 143x - 90) - (8x^2 - 48x) = -95x - 90 \]
  8. Divide \(-95x\) by \(x\) to get \(-95\).
  9. Multiply \(-95\) by \(x - 6\): \(-95x + 570\).
  10. Subtract: \[ (-95x - 90) - (-95x + 570) = -660 \]

Remainder is \(-660 \neq 0\), so \(x - 6\) is not a factor.

Conclusion

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

d. \(x - 6\).