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