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.2, 1+i, 2−√3

Answer 1

Given: Polynomial has a leading coefficient of 1

and zeroes at: x = 2, 1+i, 2-sqrt(3)

Find: Equation of the Polynomial in standard form.

Solution:

Imaginary and irrational roots come in conjugate pairs.

P = (x-2)*[x - (1+i)]*[x - (1-i)*[x - (2-sqrt(3))]*[x - (2+sqrt(3))]

P = (x-2)*[x^2 - 2x + 2]*[x^2 - 4x + 1]

P = (x-2)[x^4 -4x^3 + x^2 - 2x^3 - 2x + 4x^2 - 2x + 2x^2 - 8x + 2]

P = (x-2)[x^4 -6x^3 + 7x^2 - 12x + 2]

P = x^5 -6x^4 + 7x^3 - 12x + 12x2 + 2x - 2x^4 + 12x^3 - 14x^2 + 24x - 4

P = x^5 -8x^4 + 19x^3 -2x^2 + 26x - 4

Answer 2

(x-2) [ ( x - 1 ) + i ] [ ( x - 1 ) - i ] [ 2 - rad3 ] [ 2 + rad3 ]

(x-2) [ (x-1)^2 - (-1)] [ 4 - 3] =

(x-2) [ (x^2 - 2x + 1 + 1] =

(x-2) [ x^2 -2x + 2] =

x^3 - 2x^2 -2x^2 + 4x + 2x - 4 =

x^3 - 4x^2 + 6x - 4