solve problem

Question

Find the polynomial function of lowest degree with rational coefficients that has the given number as some of its zero?

 

4+i, 4

Answer 1

To find a polynomial of lowest degree, we'll want to find one that fits the problem's constraints with as few zeroes as possible.

 

The polynomial will include the two given zeroes: 4 and 4+i. We also have to include the complex zero's conjugate, 4-i, since complex zeroes always come in pairs. We needn't add any other zeroes.

 

The polynomial we're looking for will have one factor that gives each of those three zeroes:

 

(x - 4)(x - (4+i))(x - (4-i))

(x - 4)(x² - (4-i)x  - (4+i)x + (4+i)(4-i))

(x - 4)(x² - 4x + ix - 4x - ix + 16 - 4i + 4i + 1)

(x - 4)(x² - 8x + 17)

(x³ - 8x² + 17x - 4x² + 32x - 68)

x³ - 12x² + 49x - 68