Construct a polynomial function with the state properties. Third degree, with zeros 4-i, 4+i, and 5, and a leading coefficient of -5

Answer 1

Jerome, this is how to tackle this:

With the given zeros, we can write:

P(x) = (x - [4 - i]) (x - [4 + i]) (x - 5)

= (x - 4 + i) (x - 4 - i) (x - 5)

= ([x - 4] + i) ([x - 4] - i) (x - 5)

= ([x - 4]² - i²) (x - 5) using the difference of 2 squares formula (a - b) (a + b) = (a - b)²

= ([x - 4]² + 1) (x - 5)

= (x² - 8x + 17) (x - 5)

= x³ - 13 x² + 57x - 85 . . . . . . . . . . Eqn (1)

The question requires a leading coefficient of -5 so just multiply through by -5 which will have the same zeros as Eqn (1).

So, desired polynomial is -5x³ + 65x² - 171x + 255