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?

Rationale

P(x) could be (x-3)(x²+2).

Given that the polynomial P(x) has integer coefficients and includes the roots 3 and i√2, the conjugate root theorem necessitates that -i√2 is also a root. This means that the polynomial can be expressed as (x - 3)(x - i√2)(x + i√2) = (x - 3)(x² + 2), which matches option C.

A ((x-3)(x-2)^2) YOUR ANSWER

This polynomial has roots at x = 3 and x = 2, with 2 being a double root. It does not account for the complex roots i√2 and -i√2, which are required for the polynomial to maintain integer coefficients while having the given roots.

B ((x-3)(x²-2)) YOUR ANSWER

While this option contains the root x = 3, the quadratic factor x² - 2 has roots that are not complex conjugates (specifically, √2 and -√2). As a result, this fails to include the necessary complex roots i√2 and -i√2, making it an incorrect choice.

C ((x-3)(x²+2)) CORRECT

This polynomial correctly captures the required roots. It includes x = 3 and also the roots of the quadratic factor x² + 2, which are i√2 and -i√2. Therefore, it adheres to the stipulations of having integer coefficients and the specified roots.

D ((x+3)(x²-2)) YOUR ANSWER

This option introduces a root at x = -3, which is not part of the given roots of the polynomial. Additionally, the quadratic factor x² - 2 does not yield the necessary complex roots, thus making this choice invalid.

E ((x+3)(x^2+2)) YOUR ANSWER

Similar to option D, this polynomial includes a root at x = -3. The quadratic x² + 2 does yield the necessary complex roots i√2 and -i√2, but the presence of -3 as a root disqualifies this option since it does not match the required roots of the polynomial.

Conclusion

The requirements for the polynomial P(x) include having integer coefficients and specific roots of 3 and i√2, necessitating the inclusion of -i√2 as well. Among the given choices, only (x-3)(x²+2) fulfills these criteria, making it the correct polynomial expression. All other options either introduce extraneous roots or do not accommodate the required complex conjugate roots.

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?

P(x) could be (x-3)(x²+2).

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