Double Root
Double Root
A root of a polynomial equation with multiplicity 2. Also refers to a zero of a polynomial function with multiplicity 2.
See also
Key Formula
f(x)=a(x−r)2⋅q(x)
Where:
- r = The double root — the value of x where the polynomial equals zero with multiplicity 2
- a = The leading coefficient (a nonzero constant)
- q(x) = The remaining polynomial factor after the squared factor is extracted
- (x−r)2 = The squared factor that produces the double root at x = r
Worked Example
Problem: Find the roots of f(x) = x² − 6x + 9 and determine whether any root is a double root.
Step 1: Factor the quadratic. Recognize this as a perfect square trinomial.
x2−6x+9=(x−3)2
Step 2: Set the factored form equal to zero and solve.
(x−3)2=0⟹x−3=0⟹x=3
Step 3: Identify the multiplicity. The factor (x − 3) appears twice (exponent 2), so x = 3 is a root with multiplicity 2.
Multiplicity of x=3 is 2
Step 4: Verify using the discriminant. For ax² + bx + c, the discriminant is b² − 4ac.
Δ=(−6)2−4(1)(9)=36−36=0
Step 5: A discriminant of zero confirms exactly one distinct root with multiplicity 2 — a double root.
Answer: x = 3 is a double root of f(x) = x² − 6x + 9.
Another Example
This example shows a double root inside a higher-degree polynomial (cubic), where not every root is a double root. It also demonstrates how to find a double root using synthetic division rather than the discriminant.
Problem: Find all roots of g(x) = x³ − 5x² + 8x − 4 and identify any double roots.
Step 1: Test possible rational roots. Try x = 1: g(1) = 1 − 5 + 8 − 4 = 0. So x = 1 is a root.
g(1)=1−5+8−4=0
Step 2: Divide g(x) by (x − 1) using synthetic or polynomial division.
x3−5x2+8x−4=(x−1)(x2−4x+4)
Step 3: Factor the remaining quadratic. Notice it is a perfect square trinomial.
x2−4x+4=(x−2)2
Step 4: Write the complete factorization and list all roots with their multiplicities.
g(x)=(x−1)(x−2)2
Answer: x = 1 is a simple root (multiplicity 1) and x = 2 is a double root (multiplicity 2).
Frequently Asked Questions
What does a double root look like on a graph?
At a double root, the graph of the polynomial touches the x-axis but does not cross it. The curve reaches the axis, bounces off, and turns back in the direction it came from. This creates a tangent point on the x-axis, unlike a simple root where the graph passes straight through.
How do you know if a quadratic has a double root?
For a quadratic equation ax² + bx + c = 0, compute the discriminant Δ = b² − 4ac. If Δ = 0, the quadratic has exactly one distinct root with multiplicity 2 — a double root. If Δ > 0, there are two distinct real roots, and if Δ < 0, there are no real roots.
What is the difference between a double root and a repeated root?
They mean the same thing when the repetition count is two. A double root is a root repeated exactly twice (multiplicity 2). The broader term 'repeated root' can refer to any root with multiplicity greater than 1, including triple roots (multiplicity 3) and beyond.
Double Root vs. Triple Root
| Double Root | Triple Root | |
|---|---|---|
| Definition | A root with multiplicity 2 | A root with multiplicity 3 |
| Factor form | Contains (x − r)² | Contains (x − r)³ |
| Graph behavior at root | Touches the x-axis and bounces back (does not cross) | Crosses the x-axis with an inflection point (flattens as it crosses) |
| Minimum polynomial degree | Degree 2 (quadratic) | Degree 3 (cubic) |
| Discriminant signal (quadratic) | Δ = 0 confirms a double root | Not applicable to quadratics |
Why It Matters
Double roots appear frequently in quadratic equations, especially when you use the discriminant to classify solutions. In physics and engineering, a double root of a characteristic equation signals critical damping — the boundary between oscillating and non-oscillating behavior. Understanding double roots also helps you sketch accurate polynomial graphs, since the curve behaves differently at a double root (tangent) than at a simple root (crossing).
Common Mistakes
Mistake: Counting a double root as two separate roots when listing distinct solutions.
Correction: A double root gives only one distinct x-value. The '2' refers to its multiplicity (how many times the factor repeats), not to two different numbers. When asked for distinct roots, list x = r once and note its multiplicity.
Mistake: Assuming the graph crosses the x-axis at a double root.
Correction: At a double root the graph touches the x-axis and turns back — it does not cross. Only roots with odd multiplicity (1, 3, 5, …) produce a crossing. Sketch the curve carefully at points of even multiplicity.
Related Terms
- Root — General term for a solution of a polynomial equation
- Polynomial — The type of expression that has double roots
- Multiplicity — The count of how many times a root repeats
- Zero of a Function — An x-value where the function equals zero
- Triple Root — A root with multiplicity 3 instead of 2
- Equation — The statement set to zero to find roots
- Discriminant — Used to detect a double root when Δ = 0