Final answer:
A polynomial with rational coefficients that has √8 and 3i as zeros is P(x) = x⁴ + x² - 72, obtained by pairing each zero with its conjugate and then multiplying the corresponding factors.
Step-by-step explanation:
To find a polynomial function of the lowest degree with rational coefficients that has the given numbers as zeros, we first need to consider that complex and irrational roots come in conjugate pairs.
Since √8 is a root, its conjugate, -√8, must also be a root. Similarly, because 3i is a root, its conjugate, -3i, must also be a root.
The zeros of the polynomial are √8, -√8, 3i, and -3i. To create the polynomial, we form factors from these zeros:
- For √8 and -√8, the factor is (x-√8)(x+√8) which simplifies to x²-8.
- For 3i and -3i, the factor is (x-3i)(x+3i) which simplifies to x²+9.
Multiplying these factors together gives us the polynomial:
P(x) = (x²-8)(x²+9)
The polynomial function of the lowest degree with rational coefficients that has √8 and 3i as zeros is P(x) = x⁴ + x² - 72.