Final answer:
To create a degree 4 polynomial with given zeros, we form factors for each zero including conjugate pairs for complex zeros. Multiplying these factors and expanding them gives us the required polynomial f(x) in the form ax^4 + bx^3 + cx^2 + dx + e.
Step-by-step explanation:
To form a polynomial f(x) with real coefficients of degree 4 and zeros 5, 3i, and 2 (with multiplicity 2), we need to consider the following:
- For the real zero at x = 5, the factor is (x - 5).
- Since the coefficients must be real and we have a complex zero 3i, its conjugate -3i is also a zero. Therefore, the factors for these zeros are (x - 3i) and (x + 3i).
- The zero at x = 2 with multiplicity 2 implies the factor (x - 2)².
Multiplying these factors together gives us the polynomial:
f(x) = (x - 5)(x - 3i)(x + 3i)(x - 2)²
Expanding the factors that include the complex zeros, we get:
f(x) = (x - 5)((x - 3i)(x + 3i))(x - 2)²
f(x) = (x - 5)(x² + 9)(x - 2)²
Finally, we can expand all factors to write f(x) as a polynomial of degree 4:
f(x) = (x² + 9)(x - 5)(x - 2)(x - 2)
This step may involve a bit of algebra to multiply out the terms and combine like terms to express f(x) in standard polynomial form ax⁴ + bx³ + cx² + dx + e.