Question

Find a degree 3 polynomial with real coefficients having zeros 1

and 2−2i and a lead coefficient of 1. Write P in expanded form. Be sure to write the full equation, including P(x)=

Answer 1

9514 1404 393

Answer:

P(x) = x³ -5x² +12x -8

Explanation:

If the coefficients are real, then the complex roots must be conjugates. The third root is 2+2i. For root r, (x -r) is a factor, so the factorization is ...

P(x) = (x -1)(x -2 +2i)(x -2 -2i) = (x -1)((x -2)² +4) = (x -1)(x^2 -4x +8)

Expanding further, we find ...

P(x) = x³ -5x² +12x -8