Hi Dave,

Your graph may indicate there are no real roots but there are complex roots.

Lets look at a quadratic first. If a quadratic has roots s and t (whether real or complex) then it can be written

(x - s)(x - t)

Expansion gives

(x - s)(x - t) = x² - (s + t) x + st.

Hence the coefficient of x is minus the sum of the roots. Hence , for example for the quadratic equation x² + 2x + 5 = 0 the sum of the roots is -2.

What about a quartic? If a quartic equation has roots s, t, u and v then he quartic can be written

(x - s)(x - t)(x - u)(x - v) = 0.

Expand this. Where do you see s + t + u + v?

I hope this helps,
Penny