Final answer:
To deduce the polynomial with the given roots of 5; 3 + 2i; and 1 - 3i, we create factors from each root and also include the conjugate complex roots 3 - 2i and 1 + 3i. By multiplying the factors (x - root) together, we obtain a polynomial with real coefficients.
Step-by-step explanation:
To deduce the polynomial with the given roots of 5; 3 + 2i; and 1 - 3i, we need to create factors from each of these roots. Due to the complex conjugate root theorem, the polynomial will have the roots 3 - 2i and 1 + 3i as well. Each root r corresponds to a factor of the form (x - r). Therefore, the polynomial will be the product of these factors:
- (x - 5)
- (x - (3 + 2i))
- (x - (3 - 2i))
- (x - (1 - 3i))
- (x - (1 + 3i))
To find the polynomial, we multiply these factors out:
p(x) = (x - 5) × (x - (3 + 2i)) × (x - (3 - 2i)) × (x - (1 - 3i)) × (x - (1 + 3i)).
This multiplication will give us a polynomial with real coefficients because the complex roots come in conjugate pairs and their product will result in a polynomial with real coefficients.