Final answer:
To write a degree 5 polynomial with given roots 1, 2i, and 3i, include their conjugates -2i and -3i and multiply the corresponding factors to get f(x) = (x - 1)(x2 + 4)(x2 + 9).
Step-by-step explanation:
To write the equation of a polynomial function of degree 5 with roots 1, 2i, and 3i, we also need the conjugate pairs of the imaginary roots for a polynomial with real coefficients. Since complex roots always come in conjugate pairs, the roots will be 1, 2i, -2i, 3i, and -3i.
Each root x=a corresponds to a factor of the form (x-a). Hence, from the given roots, we can write the factors:
(x-1) for the root 1,
(x-2i) and (x+2i) for the roots 2i and -2i, and
(x-3i) and (x+3i) for the roots 3i and -3i.
The polynomial is the product of all these factors. Multiplying these factors out, we would get the equation of the polynomial:
- (x - 1)
- (x - 2i)(x + 2i)
- (x - 3i)(x + 3i)
When we multiply the pairs of conjugates, we get real-number quadratic factors:
- (x2 + 4) from (x - 2i)(x + 2i)
- (x2 + 9) from (x - 3i)(x + 3i)
Thus, the polynomial equation is:
f(x) = (x - 1)(x2 + 4)(x2 + 9)