Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.
3, -13, and 5 + 4i
f(x) = x4 - 8x3 - 12x2 + 400x - 1599
f(x) = x4 - 200x2 + 800x - 1599
f(x) = x4 - 98x2 + 800x - 1599
f(x) = x4 - 8x3 + 12x2 - 400x + 1599

Answer 1

Answer:

third option

Step-by-step explanation:

Given f(x) has zeros x = a, x = b, then

the factors are (x - a), (x - b) and the polynomial is the product of the factors.

Complex zeros occur as conjugate pairs, thus

5 + 4i is a zero then 5 - 4i is a zero

The zeros are x = 3, x = - 13, x = 5 + 4i, x = 5 - 4i, thus the factors are

(x - 3), (x - (- 13)), (x - (5 + 4i)), (x - (5 - 4i)), that is

(x - 3), (x + 13), (x - 5 - 4i), (x - 5 + 4i)

= (x - 3), (x + 13), ((x - 5) - 4i), ((x - 5) + 4i)

The polynomial is the product of the factors

f(x) = (x - 3)(x + 13)((x - 5) - 4i)((x - 5) + 4i) ← expand factors in pairs

     = (x² + 10x - 39)((x - 5)² - 16i²) → i² = - 1

     = (x² + 10x - 39)(x² - 10x + 25 + 16)

     = (x² + 10x - 39)(x² - 10x + 41) ← distribute

     = $x^{4}$ - 10x³ + 41x² + 10x³ - 100x² + 410x - 39x² + 390x - 1599 ← collect terms

     = $x^{4}$ - 98x² + 800x - 1599