Math CentralQuandaries & Queries


Question from dave, a student:

This is a algebra problem that i am confused about:
The sum of the roots of x^4-x^3+5x^2+4=0 is:
i tried graphing it, but it shows that there are no roots, but the answer is 1. are they wrong?

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,

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