Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the zeros -1 and 4-i. Write the function in standard form

Answer 1

Answer:

  f(x) = x^3 -7x^2 +9x +17

Step-by-step explanation:

If the polynomial has real coefficients, then any complex roots come in conjugate pairs. Since 4-i is a root, 4+i is also a root. For each root r, the polynomial has a factor (x-r), so the minimal polynomial is ...

  f(x) = (x -(-1))(x -(4-i))(x -(4+i))

  = (x +1)((x -4)^2 -i^2) . . . . difference of squares

  = (x +1)(x^2 -8x +17)

  f(x) = x^3 -7x^2 +9x +17