Given Some Zeros Of A Polynomial Function Find All Zeros Of The Function

P(x) = 2x^4 – x^3 – 27x^2 + 16x – 80, with zeros: 4, -4

Solution

---

The easiest way to do this is to use synthetic division to divide the factors associated with the zeros 4 and -4 into the polynomial then to solve the resulting quadratic equation by factoring or by the quadratic formula. I’m unsure if you’ve done synthetic division or not. If not, you can use long division.

Synthetic Division:

_________________

4 | 2 -1 -27 16 -80

___8__28__4___80_

2 7 1 20 0

And then

____________

-4 | 2 7 1 20

____-8__4__-20

2 -1 5 0

So the resulting quotient of (2×4 – x3 – 27×2 + 16x – 80)/[(x – 4)(x + 4)] is 2×2 – x + 5. It looks like this is not factorable so using the quadratic formula we get: x = [1 ± √(1 – 4(2)(5)]/[(2)(2)] = (1 ± √39i)/4