# Thread: Find All Zeros (real and imaginary)

1. ## Find All Zeros (real and imaginary)

I need to use the given zero to find all the zeros real or imaginary of the function and also write f as a product of linear factors.

Zero: 5-i

F(x)=(3x^6)-(43x^5)+(218x^4)-(428x^3)+(156x^2)+300x-104

2. Originally Posted by Kairo
I need to use the given zero to find all the zeros real or imaginary of the function and also write f as a product of linear factors.

Zero: 5-i

F(x)={3x^6}-{43x^5}+{218x^4}-{428x^3}+{156x^2}+300x-104
If $\displaystyle 5-i$ is a root of $\displaystyle f(x)$, then $\displaystyle 5+i$ also a root. Thus $\displaystyle \left[z-(5-i)\right]$ and $\displaystyle \left[z+(5-i)\right]$ are factors of $\displaystyle f(x)$. Divide $\displaystyle f(x)$ by $\displaystyle x^2-10 x+26$, and then you will get the quadric $\displaystyle 3x^4-13x^3+10 x^2+10x-4.$ By plugging in possible integer roots, find that $\displaystyle x = 2$ yields the quadric zero, so $\displaystyle x-2$ is a factor of it. Divide the quadric by $\displaystyle x-2$, and you will get $\displaystyle 3x^3-7x^2-4x+2$. Keep going like this and find the roots of the cubic. Writing the equation as the product of linear factors means writing $\displaystyle f(x)$ as $\displaystyle (x-a)(x-b)(x-c)(x-d)(x-e)(x-f)$, where $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, $\displaystyle d$, $\displaystyle e$, and $\displaystyle f$ are the roots you were asked to find, of which we already have three: $\displaystyle 5-i$, $\displaystyle 5+i$, and $\displaystyle 2$, and the other three are the roots of the cubic $\displaystyle 3x^3-7x^2-4x+2 = 0$.