Question

Find a degree 3 polynomial with real coefficients having zeros 3 and 1 − 4i and a lead coefficient of 1. Write P in expanded form.

Answer 1

`\text{Let,}\\\\~~~~~~~x=1-4i,\\\\\implies x-1 = -4i\\ \\\implies (x-1)^2 =(-4i)^2\\\\\implies x^2 -2x +1 = -16\\\\\implies x^2 -2x +1 +16 = 0\\\\\implies x^2 -2x +17=0\\\\\text{So,}~ x^2 -2x +17 ~\text{is a factor of the 3 degree polynomial.}\\ \\\text{The polynomial is,}\\\\P(x) = (x-3)(x^2 -2x +17)\\\\~~~~~~~=x^3-2x^2+17x -3x^2 +6x -51\\\\~~~~~~~=x^3 -5x^2 +23x -51`