Polynomial Functions

Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. Write the polynomial in standard form.

3, 4+i

Answer 1

A polynomial with complex roots has a degree of 2.  But since we have one real root, and a complex root that has a conjugate:

 

x₁ = 3

x₂ = 4 + i

x₃ = 4 - i

 

 

The degree of the polynomial will be 3 because there are 3 roots. So the polynomial is

 

f(x) = (x - 3)[x + (-4 - i)][x + (-4 + i)]

 

f(x) = (x - 3)[x² + x(-4 + i) + x(-4 - i) + (16 - i²)]

 

f(x) = (x - 3)[x² - 4x + ix - 4x - ix + 17]

 

f(x) = (x - 3)(x² - 8x + 17)

 

f(x) = x(x² - 8x + 17) - 3(x² - 8x + 17)

 

f(x) = x³ - 8x² + 17x - 3x² + 24x - 51

 

f(x) = x³ - 11x² + 41x - 51