Given that 6i is a zero of g , write the polynomial in factored form as a product of linear factors: g ( c ) = c^6-8c^5+41c^4-256c^3+144c^2+1152c-1269

Answer 1

use desmos graphing calculator and the zeros are about -2, 1.5, 2 and 6.7 so the factors would be about

(x-6i)(x+6i)(x-1.5)(x-2)(x+2)(x-6.7)

2 imaginary factors and 4 real factors

or there might be a problem in the problem:

if 6i is a polynomial factor so is -6i. imaginary factors always come in conjugate pairs

(c-6i)(c+6i) = (c^2+36) as a quadratic factor

divide that factor into the 6 degree polynomial to get a 4th degree polynomial. but it doesn't come out even

Maybe the 6th degree polynomial was miscopied or misprinted, and should have had no linear term, and wtih a positive constant term. Then divide by c^2+36 which

gives a 4th degree polynomial = c^4-8c^3+5c^2-4c +144

it has only 2 real solutions: c=about 3.48153 and 6.92878

so the corresponding factors are (x-3.48153)(x-6.92878)

the last 2 factors are more imaginary factors, found with the solutions of a quadratic equation which is about c^2+2.41032c+5.96944 = 0

the discriminant <0, so the solutions are not real

use the quadratic formula

x = -1.20516 +/-[sqr(2.41032^2 -4(5.96944))]/2

the two imaginary factors are

about (x+1.2+2.13i)(x+1.2-2.13i)

multiply all the factors together

g(x) = approximately

(x-6i)(x+6i)(x+1.2+2.13i)(x+1.2-2.13i)(x-3.48153)(x-6.9944)

4 imaginary factors, 2 real factors

Odds are there's a mistake in the problem, and maybe a mistake in the above calculations

You could also try graphing the polynomial in a graphing calculator, the x intercepts are the zeros. Then change their signs and stick an x in front of them to get the corresponding factors

or use an online equation solver. They seem to give two irrational zeros, leaving 4 imaginary zeros