Repeated Roots

Consider the quadratic x^2 - 4x + 4 = 0. Factor it: (x - 2)^2 = 0. The only solution is x = 2, but the factor (x - 2) appears twice. This is different from x^2 - 5x + 6 = 0, which factors as (x - 2)(x - 3) = 0 and has two distinct roots.

The distinction matters. When you count roots "with multiplicity," both quadratics have two roots — the first has root 2 with multiplicity 2, and the second has roots 2 and 3 each with multiplicity 1. But the geometry is completely different: the parabola y = (x - 2)^2 touches the axis at x = 2 and bounces back up, while y = (x - 2)(x - 3) cuts through the axis at two points.

This touching-versus-crossing distinction is not a coincidence unique to parabolas. It persists at every degree, and it leads to a beautiful connection between roots of a polynomial and roots of its derivative.

What "repeated root" means precisely

For the polynomial f(x) = (x - 1)^3 (x + 2)^2 (x - 5), the root x = 1 has multiplicity 3, the root x = -2 has multiplicity 2, and the root x = 5 has multiplicity 1. The total count — 3 + 2 + 1 = 6 — equals the degree of f.

This always holds: for any polynomial of degree n with real (or complex) coefficients, the multiplicities of all roots add up to exactly n. This is a consequence of the Fundamental Theorem of Algebra.

Geometry of repeated roots

The multiplicity determines how the curve behaves at the root.

Multiplicity 1 (simple root). The curve crosses the axis at a nonzero angle — it passes straight through.

Multiplicity 2 (double root). The curve touches the axis and turns back. It does not cross — the sign of f(x) does not change. Think of a ball bouncing off the floor.

Multiplicity 3 (triple root). The curve crosses the axis, but with a flat, inflection-like shape — the curve is tangent to the axis at the root, lingers near zero, and then continues through. The sign does change, but gently.

The pattern: even multiplicity means the curve touches and turns back (no sign change). Odd multiplicity means the curve crosses (sign changes), but for multiplicities greater than 1, the crossing is flattened.

The condition for repeated roots

Here is the central question: given a polynomial f(x), how do you tell whether it has a repeated root without fully factoring it? The answer involves the derivative.

If r is a root of multiplicity k \geq 2, then f(x) = (x - r)^k \cdot g(x) for some polynomial g with g(r) \neq 0. Differentiate using the product rule:

At x = r: the factor (x - r)^{k-1} becomes 0 (since k \geq 2 means k - 1 \geq 1). So f'(r) = 0.

Conversely, if r is a simple root (multiplicity 1), then f(x) = (x - r) \cdot g(x) with g(r) \neq 0, and f'(r) = 1 \cdot g(r) + 0 = g(r) \neq 0.

This gives a clean test:

This is remarkable. The derivative — a concept from calculus — detects repeated roots purely algebraically. You do not need to fully factor f(x) or find all roots. You just need the GCD of f(x) and f'(x).

Finding repeated roots

The condition gives a practical algorithm:

1. Compute f'(x).
2. Find \gcd(f(x), f'(x)) using the Euclidean algorithm for polynomials.
3. If the GCD is a constant (degree 0), there are no repeated roots.
4. If the GCD has positive degree, its roots are exactly the repeated roots of f(x).

Take f(x) = x^3 - 3x + 2. Compute f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x - 1)(x + 1).

Check f(1) = 1 - 3 + 2 = 0 and f'(1) = 3 - 3 = 0. Both are zero, so x = 1 is a repeated root.

Check f(-1) = -1 + 3 + 2 = 4 \neq 0. So x = -1 is a root of f' but not of f — it is not relevant.

Factor f(x): since (x - 1) is a repeated factor, divide: x^3 - 3x + 2 = (x - 1)^2(x + 2). The root x = 1 has multiplicity 2 and x = -2 has multiplicity 1.

The discriminant connection

For a quadratic ax^2 + bx + c = 0, you already know the discriminant D = b^2 - 4ac. When D = 0, the two roots coincide — the quadratic has a repeated root. This is the quadratic's version of the "both f and f' vanish" condition.

The discriminant of a cubic or higher-degree polynomial is a more complex expression, but it plays the same role: when the discriminant is zero, the polynomial has at least one repeated root.

For the general quadratic, the repeated root when D = 0 is x = -b/(2a). For the cubic x^3 + px + q = 0, the discriminant is \Delta = -4p^3 - 27q^2, and \Delta = 0 signals a repeated root.

Connection with derivatives: a preview

The fact that f(r) = 0 and f'(r) = 0 at a double root is the beginning of a deeper pattern. If r has multiplicity k, then:

and f^{(k)}(r) \neq 0. Each successive derivative "peels off" one layer of the repeated factor. The first nonzero derivative tells you the multiplicity.

For a triple root: f(r) = 0, f'(r) = 0, f''(r) = 0, but f'''(r) \neq 0.

This makes sense geometrically. At a simple root, the curve crosses the axis with a nonzero slope — f'(r) \neq 0, so the tangent line is tilted. At a double root, the slope is zero — the curve is flat, tangent to the axis. At a triple root, even the curvature (f'') is zero — the curve is not just tangent but has an inflection point on the axis.

You will study derivatives properly in calculus. For now, the key takeaway is this: the derivative of a polynomial is itself a polynomial, computable by the rule that \frac{d}{dx}(x^n) = nx^{n-1}, and it reveals the multiplicity structure of the roots.

Two worked examples

Common confusions

• "A repeated root means the equation has fewer solutions." It depends on how you count. If you count distinct solutions, yes — x^2 - 4x + 4 = 0 has one distinct solution. But if you count with multiplicity, it has two: the root 2 counted twice. Most theorems in algebra use the multiplicity count, because it makes the accounting clean.

• "If the discriminant is zero, the polynomial has a double root." The discriminant being zero means there is at least one repeated root, but the multiplicity could be higher than 2. For (x - 1)^3 = x^3 - 3x^2 + 3x - 1, the discriminant is zero, but x = 1 is a triple root, not a double root.

• "You need calculus to find repeated roots." You do not need the full machinery of calculus — you only need the derivative formula for polynomials, which is the power rule \frac{d}{dx}(x^n) = nx^{n-1}. This formula can be stated and used purely algebraically, without limits or continuity. In fact, the "formal derivative" of a polynomial is defined in abstract algebra without any reference to calculus at all.

• "If the curve touches the axis, the root has multiplicity exactly 2." The curve also touches (without crossing) at multiplicity 4, 6, or any even multiplicity. And it crosses with a flat shape at multiplicity 3, 5, or any odd multiplicity greater than 1. The key is even versus odd: even multiplicities touch, odd multiplicities cross.

• "(x - r)^2 divides f(x) means r is a root of f'(x)." This is true — but the converse is not. A root of f'(x) is not necessarily a repeated root of f(x); it could be a local maximum or minimum of f that happens not to be on the axis. You need both conditions: f(r) = 0 and f'(r) = 0.

Going deeper

The GCD method in detail

The polynomial GCD \gcd(f(x), f'(x)) captures all the repeated-root information of f(x). If f(x) = (x - r_1)^{m_1} (x - r_2)^{m_2} \cdots (x - r_s)^{m_s} with each m_i \geq 1, then

The GCD includes only the roots with multiplicity \geq 2, and each such root appears with multiplicity reduced by one. So if f has a triple root at r, the GCD has a double root at r. If f has a double root, the GCD has a simple root.

To extract all repeated roots, divide f(x) by \gcd(f(x), f'(x)). The result is the square-free part of f — a polynomial with the same roots as f, but all with multiplicity 1.

Formal derivatives

In abstract algebra, the derivative of a polynomial is defined without any reference to limits. For f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, the formal derivative is

This is a purely algebraic operation — it works over any field, not just the real numbers. The product rule (fg)' = f'g + fg' holds for formal derivatives, and that is all you need for the repeated-root theory. The connection to tangent lines and rates of change is a consequence of this algebraic definition when you specialise to real-valued functions, not a prerequisite for it.

Multiplicity and Taylor expansion

If r is a root of multiplicity k, the Taylor expansion of f(x) around x = r starts at the k-th power:

The first k - 1 derivatives vanish, so the first k - 1 terms of the Taylor series are zero. The leading term is (x - r)^k times a nonzero constant. This is why the curve is "flat to order k" at the root — the local behaviour is dominated by (x - r)^k, which vanishes faster and faster as k increases.

Where this leads next

Repeated roots connect algebra and calculus in ways that deepen as you study both subjects.

• Polynomial Equations — the general theory of roots, including the Fundamental Theorem of Algebra and root counting.
• Discriminant and Nature of Roots — the discriminant as the universal detector of repeated roots, with detailed treatment for quadratics.
• Higher Degree Equations — Vieta's formulas and transformations for equations of any degree.
• Cubic Equations — the degree-three case, where the cubic discriminant \Delta = 0 signals a repeated root.
• Derivative — the full calculus treatment of the derivative, which gives the repeated-root condition its geometric meaning.