A repeated root of a polynomial is a root that appears more than once in the polynomial’s factorization. If you can factor a polynomial and the same root shows up two, three, or more times, that root is “repeated.” The number of times it appears is called its multiplicity. A root with multiplicity 2 is commonly called a double root, multiplicity 3 is a triple root, and so on.
How Repeated Roots Work
Every polynomial can be broken down into factors. When you factor completely, each factor of the form (x − r) corresponds to a root at x = r. If that same factor appears more than once, you have a repeated root.
Take the polynomial x² + 6x + 9. It factors into (x + 3)(x + 3), which simplifies to (x + 3)². The root here is x = −3, and because the factor appears twice, it’s a repeated root with multiplicity 2. Compare that to something like (x − 1)(x − 4), where x = 1 and x = 4 are each simple roots with multiplicity 1.
The formal definition: if (x − r) raised to the power m divides a polynomial, but (x − r) raised to the power m + 1 does not, then r is a root of multiplicity m. When m is 1, it’s a simple root. When m is 2 or greater, it’s a repeated root.
How to Spot a Repeated Root in a Quadratic
For quadratic equations (ax² + bx + c = 0), the quickest test is the discriminant: b² − 4ac. When this value equals zero, the equation has exactly one repeated real root (a double root). A positive discriminant means two distinct real roots, and a negative one means no real roots at all.
For example, in x² + 6x + 9 = 0, the discriminant is 6² − 4(1)(9) = 36 − 36 = 0. That zero result confirms the double root at x = −3.
Counting Roots With Multiplicity
The fundamental theorem of algebra says that every polynomial of degree n has exactly n roots (counting complex numbers), as long as you count each root according to its multiplicity. So a degree-4 polynomial always has exactly 4 roots when you count this way, even if some of those roots are the same value repeated.
Consider (x − 2)³(x + 5). This is a degree-4 polynomial. It has root x = 2 with multiplicity 3 and root x = −5 with multiplicity 1. That’s 3 + 1 = 4 roots total, matching the degree. Without counting multiplicity, you’d say it has only two distinct roots, which obscures the polynomial’s full structure.
What Repeated Roots Look Like on a Graph
Repeated roots change how a polynomial’s graph interacts with the x-axis, and the behavior depends on whether the multiplicity is even or odd.
At a root with even multiplicity (2, 4, 6, …), the graph touches the x-axis and bounces back without crossing it. Picture a ball dropping onto a floor and bouncing up. The curve comes down to the axis, just barely makes contact at that point, and turns back the way it came. A classic example is y = (x − 3)², which touches the x-axis at x = 3 and bounces off.
At a root with odd multiplicity greater than 1 (3, 5, 7, …), the graph still crosses the x-axis, but it flattens out near the crossing point. Instead of cutting cleanly through the axis like a simple root would, the curve slows down, flattens, and then passes through. The higher the multiplicity, the flatter the curve looks at that crossing.
This visual pattern is useful in reverse, too. If you’re looking at a graph and see it bounce off the x-axis at some point, you know that point is a root with even multiplicity. If it crosses but looks flat near the axis, it’s likely an odd multiplicity of 3 or higher.
The Derivative Connection
There’s a clean relationship between repeated roots and derivatives. If a value r is a repeated root of a polynomial f(x), then plugging r into both f(x) and its derivative f′(x) gives zero. In other words, a repeated root is always also a root of the derivative.
This works in both directions: r is a repeated root of f(x) if and only if f(r) = 0 and f′(r) = 0. If f(r) = 0 but f′(r) is not zero, then r is a simple root. This gives you a computational way to test whether a root you’ve found is repeated, without fully factoring the polynomial.
The geometric intuition here is straightforward. At a repeated root, the graph either bounces off the x-axis or flattens as it crosses. In both cases, the slope of the curve at that point is zero, which is exactly what f′(r) = 0 means.
Repeated Roots in Differential Equations
If you’re encountering repeated roots in a course on differential equations, the concept takes on a slightly different role. When solving a second-order linear differential equation with constant coefficients, you set up what’s called a characteristic equation, which is a quadratic. The roots of that quadratic determine the form of your solution.
When the characteristic equation has two distinct roots, you get two naturally independent solution pieces. But when it has a repeated root r, both pieces would normally give you the same function, which isn’t enough to build a complete solution. The fix is to multiply one of the pieces by t (the independent variable). So instead of two identical terms, your general solution becomes c₁e^(rt) + c₂te^(rt). That extra factor of t is what makes the two parts independent and gives you the flexibility to satisfy initial conditions.
This pattern extends to higher-order equations. A root of multiplicity 3 would produce terms involving e^(rt), te^(rt), and t²e^(rt). Each additional repetition of the root adds another power of t as a multiplier.