22. The polynomial P(x) of degree 3 has integer coefficients. If 3 and i*√2 are two of the roots of P(x), then P(x) could be which of the following?
Answer: C
P(x) could be (x-3)(x²+2)
The polynomial P(x) can be represented as (x-3)(x²+2), which accounts for the root 3 and includes a conjugate pair for the complex root i√2, ensuring that P(x) has integer coefficients.
A) (x-3)(x-2)^2
This option does not satisfy the requirement for the roots, as it only includes the root 3 and does not account for the complex root i√2. Additionally, the other root in this polynomial, which is 2, does not align with the given roots.
B) (x-3)(x²-2)
While this option includes the root 3, the term (x²-2) would yield roots of ±√2, which does not include the required complex root i√2. Therefore, this polynomial does not fulfill the criteria for having integer coefficients while capturing the specified roots.
C) (x-3)(x²+2)
This polynomial correctly includes the root 3 and the quadratic factor (x²+2), which has roots of ±i√2, satisfying the condition for complex roots. This ensures that all roots are accounted for while maintaining integer coefficients.
D) (x+3)(x²-2)
This option features the root -3, which does not match the required root of 3. Furthermore, the term (x²-2) introduces roots of ±√2, failing to meet the requirement for the complex root i√2 and thus does not satisfy the polynomial's criteria.
E) (x+3)(x²+2)
Similar to option D, this polynomial incorrectly includes the root -3 instead of the required root 3. While it does include the quadratic term (x²+2), which has roots ±i√2, the presence of -3 makes this option invalid.
Conclusion
The polynomial (x-3)(x²+2) is definitively correct as it incorporates both the integer root 3 and the necessary complex root i√2 through its conjugate pair. All other options fail to satisfy the conditions set by the question, either by omitting necessary roots or introducing incorrect ones, thus confirming option C as the only valid representation of P(x).