Write the equation for a 4th degree polynomial function that has roots at x = 3 + i, x = -2 and x = 5. The y‑intercept is (0, 25).

Question

Write an equation for this function in factored form with real values.

Answer 1

x = (3 - i) is also zero. Because in factoed form f(x) = a(x + 2)(x - 5)(x - 3 - i)(x - 3 + i) = a(x + 2)(x - 5)((x - 3)² -

i ²) = a(x² - 3x - 10)(x² - 6x + 10); now f(0) = a(-10)(10) = 25; a = - 0.25;

f(x) = - 0.25(x⁴ - 9x³ + 18x² + 30x - 100) = - 0.25x⁴ + 2.25x³- 4.5x² - 7.5x + 25

;

Answer 2

x = 3 + i

x-3 = i

x^2 -6x + 9 = -1

x-^2 -6x +10 is a factor

(x+2)(x-5)(x^2-6x+10) = (x^2-3x-10)(x^2-6x+10) = x^4-6x^3 +60x - 100

but that has (0,-100) as the intercept. to change it to (0,25) multiply by -1/4

the polynomial then is

(-1/4)x^4 + (3/2)x^3 - 12x + 25 which has the given zeroes plus the needed x=3-i to complete a conjugate pair for rational coefficients..