What are the real zeros of the polynomial
𝑃
(
𝑥
)
=
𝑥
3
−
4
𝑥
2
−
𝑥
? Select all that apply.

Question

What are the real zeros of the polynomial 
𝑃
(
𝑥
)
=
𝑥
3
−
4
𝑥
2
−
𝑥
? Select all that apply.

BlackTom AI · Accepted Answer

The problem asks for the real zeros of P(x) = x^3 − 4x^2 − x and allows selecting all that apply.
Option A: x = 2 + 5. This is x = 7, which is not a root of P(x) since P(7) ≠ 0 (the actual real zeros are 0 and 2 ± √5). Therefore this option is incorrect.
Option B: This polynomial only has imaginary zeros. That statement is false because P(x) factors as x(x^2 − 4x − 1). The cubic clearly has a real root at x = 0 and the quadratic factor has real roots 2 ± √5, so there are real zeros. Hence this option is incorrect.
Option C: x = 0. Substituting x = 0 gives P(0) = 0, so x = 0 is indeed a real zero. This option is correct.
Option D: x = 2 − 5. This is x = −3, which is not a root of P(x) since P(−3) ≠ 0, so this option is incorrect.
In summary, only x = 0 among the given choices is a real zero; the other numerical options do not satisfy P(x) = 0 and the claim about imaginary zeros is inaccurate.

What are the real zeros of the polynomial 𝑃 ( 𝑥 ) = 𝑥 3 − 4 𝑥 2 − 𝑥 ? Select all that apply.多项选择题

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