Roots of Polynomials and Roots of Unity
Every polynomial of degree n has n complex roots, with non-real roots of real polynomials arriving in conjugate pairs, and the nth roots of unity sit evenly spaced around the unit circle.
What a learner can do afterwards
- Factor a cubic with one real root into its real root and a conjugate pair
- Find all nth roots of unity for small n and place them on the Argand plane
- State what the fundamental theorem of algebra promises and what it does not
1 · Read
The fundamental theorem of algebra promises exactly as many complex roots as the degree, counting repeats. A degree 5 polynomial owns 5, a degree 7 owns 7. It promises the count, never the values. A real cubic always hides a real root, since its odd-degree ends head opposite ways and force the graph across the axis.
Factor first and read each piece. With (x minus 2)(x squared plus 1), the linear piece gives x = 2 and the quadratic gives x squared = minus 1, so i and minus i. The three roots are 2, i, and minus i, with the non-real pair arriving as conjugates. Likewise x cubed minus 8 has the single real root 2.
The nth roots of unity solve x to the n equals 1, giving n points spaced evenly around the unit circle. The 6th roots sit at the corners of a regular hexagon, every 60 degrees. The number i qualifies as a 4th root of unity, since i squared is minus one and squared again is 1.
Balance your root list against the degree before finishing. One linear factor means one real root found, and x squared plus 1 always donates the pair i and minus i. Real polynomials pair every non-real root with its conjugate, so a lone i demands a minus i nearby.
Degree sets the headcount, conjugates pair up, and roots of unity ring the circle evenly.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.