. Create a degree 3 polynomial in the form 𝑓(𝑥)=𝑎𝑥3+𝑏𝑥2+𝑐𝑥+𝑑 such that f (-6) = 0 and f (-2 + i) = 0.

Question

. Create a degree 3 polynomial in the form 𝑓(𝑥)=𝑎𝑥3+𝑏𝑥2+𝑐𝑥+𝑑 such that f (-6) = 0 and f (-2 + i) = 0. 

Answer 1

Since f(-2+i) = 0, f(-2-i) = 0.

By the Factor Theorem, x + 6, x - (-2+i). and x - (-2-i) are factors of f(x).

f(x) = (x + 6)[x - (-2+i)][x - (-2-i)]

= (x + 6)[(x+2) - i][(x+2) + i]

= (x + 6)[(x+2)² - i²]

= (x + 6)(x² + 4x + 5) = x³ + 4x² + 5x + 6x² + 24x + 30

f(x) = x³ + 10x² + 29x + 30