form a third-degree polynomial function with real coefficents such that -9 + i and -8 are zeros

Question

 form a third degree polyomial function with real coefficients such that

-9 + i and -8 are zeros

Answer 1

Since the polynomial has real coefficients, complex roots occur in conjugate pairs.  That is, since -9+i is a root, then -9-i must also be a root.  

 

A polynomial with the desired roots is:

 

[x - (-8)][x - (-9+i)][x - (-9-i)] = (x + 8)[(x+9) - i][(x+9) + i]

 

                                                         = (x + 8)[(x+9)² - i²]

 

                                                         = (x+8)(x² + 18x + 81 + 1)

 

                                                         = (x+8)(x²+18x+82)

 

                                                         = x³+18x²+82x+8x²+144x+656

 

                                                         = x³ + 26x² + 226x + 656