Complex Zeros Of Polynomial Functions...?

Question

I need to find the Complex zeros of these polynomials...

f(x)=x^3-1
~~~~ I know that 1 is the real zero. But the simplicity of the equation is throwing me off~~~~

g(x)=x^3-8x^2+25x-26
~~~~I found 2 as a real zero and by factoring out (x-2), I end up with (x^2-6x+13). That 13 bothers me...~~~~

F(x)=3x^4-x^3-9x^2+159x-52
~~~~I found -4 and 1/3 as zeros, so I have:
(x+4)(x-1/3)(3x^2-12x+39)~~~~~

Any help would be very much appreciated.

Thank You..

Answer 1

1. The zeros are the three roots of 1. One is real (=1) and the other two are complex conjugates. Think of three spokes, equally spaced, on the unit circle in the complex plane.

2. Odd order polynomials always have at least one real root, you've found it. Use the quadratic formula to get the roots of that remaining quadratic. I avoid using the quadratic formula but it's necessary when the roots are complex. [since b^2-4ac = 36 - 4*1*13 < 0 then you know that you have two complex roots]

3. Same as above, just use quadratic formula to get the roots of that remaining quadratic.
[since b^2-4ac = 144 - 4*3*39<0, then you have two complex roots]

See, you can do it!

Answer 2

considering the fact it is an unusual degree, it could have a minimum of a million actual root I graphed it to discover the actual root at x = -5, yet you additionally can try the available rational roots as +/- aspects of eighty 5 (a million , 5 , 17 , eighty 5) by skill of man made branch: -5 | ...a million ...... 13 ......57 ...... eighty 5 ......... .........-5......-40 ......-eighty 5 ---------------- ----------------- --------- .........a million .......8 .......17 .........0 <== the rest = 0, confirming a 0 quotient: x^2 + 8x + 17 x^3 + 13x^2 + 57x + eighty 5 = (x + 5)(x^2 + 8x + 17) you could now use the quadratic formula for the 2d ingredient, or complete the sq.: x^2 + 8x + 17 = 0 x^2 + 8x = -17 x^2 + 8x + sixteen = -17 + sixteen (the place sixteen = (8 / 2)^2 (x + 4)^2 = -a million x + 4 = +/- i x = -4 +/- i the two complicated zeros are x = -4 +/- i, to pass alongside with the actual root x = -5