Complex roots of polynomials

A root of a polynomial is a number x that makes it 0. The graph of $y = p(x)$ has only real x, so it shows only the real roots, where the curve meets the x-axis. This page adds a constant c and solves $p(x) + c = 0$. Raising c lifts the graph, and once it clears the axis the real roots seem to be gone. They are still there as complex numbers $a + bi$. The main view draws every root in the complex plane: real roots on the real axis, the others above or below it. The graph underneath shares its columns, so a real root on the curve sits straight below its dot in the plane.

Take $x^2 - 2x + c$, whose roots are $1 \pm \sqrt{1 - c}$. As c rises to 1 the two real roots slide together and meet at x = 1, where the graph just touches the axis: $x^2 - 2x + 1 = (x - 1)^2$, a double root. Past c = 1 the number under the square root is negative and the roots become $1 \pm i\sqrt{c - 1}$. They leave the real line at right angles, one up and one down, and from then on the vertex's height, c − 1, is the square of their imaginary part and its x is their real part. Two real roots meet the same way wherever the graph touches the axis at a low point, and raising c turns them into a pair; at a high point, such as the cubic's at c = −2, lowering c does. The link to the vertex is a property of quadratics, though, and does not carry over in general: on the cubic at c = 3 the pair has real part 1.052, while the graph's low point next to it is at x = 1.

The coefficients here are real. Taking the complex conjugate of $p(z) + c = 0$ gives $p(\bar z) + c = 0$, so if z is a root, so is $\bar z$: non-real roots come in pairs, mirrored across the real axis and joined on the plot by a dashed line. The fundamental theorem of algebra says that a polynomial of degree n has exactly n complex roots, counting each as many times as it repeats: a double root twice, the triple root of x³ at c = 0 three times. The red key checks it. As the cubic rises from c = −4 to 4 its real roots go 1, 3, 1, yet every row of the record adds up to 3. Since pairs come two at a time, an odd degree always keeps at least one real root.

Multiplying out $(x - z_1)\cdots(x - z_n)$, the coefficient of $x^{n-1}$ is $-(z_1 + \dots + z_n)$. Moving c changes only the constant term, so the sum of the roots never moves. For $x^3 - 3x$ it is 0 at every c: when the real root moves left the pair moves right, and at c = 3, −2.104 + 2 × 1.052 = 0. With Sum of roots on, a diamond marks the mean of the roots, and it stays put through any run.