Write a polynomial of degree 3 with real coefficients that has zeros 5 and −1 + 2i

Question

Could you help me with my Pre-Cal? Thanks

Answer 1

Since the polynomial has degree 3, the polynomial has at most 3 distinct roots.

Since the polynomial has real coefficients, complex roots occur in conjugate pairs. So, since -1+2i is a root, then -1-2i is also a root.

So, the only roots are -1+2i, -1-2i, and 5.

A polynomial with these roots is (x - 5)[x - (1+2i)][x - (1-2i)]

= (x - 5)[(x-1) - 2i][(x-1) + 2i]

= (x - 5)[(x-1)² - 4i²]

= (x - 5)(x² - 2x + 5)

= x³ - 2x² + 5x - 5x² + 10x - 25 = x³ - 7x² + 15x - 25