Finding complex factors of 9z^3 - 18z^2 + 5z - 10 = 0, given z - 2 is a factor.

[tpupble]

Consider the equation 9z³-18z²+5z-10=0. Given that z-2 is a factor of the equation, determine the other two factors:

I first performed polynomial long division in order to determine the product of the 2 unknown factors. I then used the quadratic formula to determine the values of Z which produced zero, hence were roots. But, the solutions I produced does not fit the coefficient of 9 in the z³ term, instead if my factors were used the z³ term would have a coefficient of 1. Do I have to use a theorem like the conjugate root theorem, any hints appreciated.

 

[Steven G]

First comment is that you can simplify sqrt(20) to 2sqrt(5).

Consider for example 2x²-8 = 0. This is the same as 2(x²-4) = 2(x-2)(x+2)=0. Then x=2 and x=-2. So the factors must be (x-2) and (x+2). BUT (x-2)(x+2) does NOT equal 2x²-8. This is basically what happened in your case.

What to do?? If x=a and x=b are the zeros of a quadratic equation, then this quadratic equation when factored will be C(x-a)(x-b) for an appropriate value for C. In your case, C=9

This issue does not only occur for 2nd degree, but for all polynomials.