Find all zeros (real and complex) and factor the polynomial completely. a) P(x) = x^2+24x+145

Answer 1

Use the quadratic formula:

x = [-24 ± √(24² - 4(145))] / 2 = [-24 ± √(-4)] / 2 = (-24 ± 2i) / 2 = -12 ± i

Answer 2

x^2 +24x + 145 =0

complete the square

x^2 +24x + 144 = -145 +144 = -1

(x+12)^2 = -1

x+12 = +/-i

by descartes sign rule there are no sign changes so no positive real roots

also 0 or 2 negative real roots and 0 or 2 imaginary roots

turns out only roots are a conjugate pair of imaginary roots

x = -12 +/-i