One of the factors of (x3 + 1) - (x - 1) is
2023
One of the factors of (x3 + 1) - (x - 1) is
Answer: E. None of the above — CONCEPTFor a polynomial P(x), an expression is a factor exactly when polynomial division leaves remainder zero. For a linear candidate x − a, this is the…
- A.
x2 + 1
- B.
x2 - 1
- C.
x - 1
- D.
More than one of the above
- E.
None of the above
Show answer & explanation
Correct answer: E
CONCEPT
For a polynomial P(x), an expression is a factor exactly when polynomial division leaves remainder zero. For a linear candidate x − a, this is the Factor Theorem condition P(a) = 0.
APPLICATION
Simplify the expression: P(x) = (x3 + 1) − (x − 1) = x3 − x + 2.
Divide by x2 + 1: P(x) = x(x2 + 1) + (2 − 2x), so the remainder is 2 − 2x.
Divide by x2 − 1: P(x) = x(x2 − 1) + 2, so the remainder is 2.
Divide by x − 1: P(x) = (x − 1)(x2 + x) + 2, so the remainder is 2.
CROSS-CHECK
For the linear candidate x − 1, P(1) = 1 − 1 + 2 = 2, confirming the same nonzero remainder.
Therefore none of the listed algebraic expressions is a factor, so the result is “None of the above.”