Triple Root
Triple Root
A root of a polynomial equation with multiplicity 3. Also refers to a zero of a polynomial function with multiplicity 3.
See also
Key Formula
f(x)=(x−r)3⋅q(x)
Where:
- r = The triple root — the value of x where the polynomial equals zero with multiplicity 3
- q(x) = The remaining polynomial factor, where q(r) ≠ 0 (so r is not a root of q)
- f(x) = The original polynomial function
Worked Example
Problem: Show that x = 2 is a triple root of the polynomial f(x) = x³ − 6x² + 12x − 8.
Step 1: Check whether x = 2 is a root by substituting into f(x).
f(2)=(2)3−6(2)2+12(2)−8=8−24+24−8=0
Step 2: Factor the polynomial completely. Recognize that x³ − 6x² + 12x − 8 matches the pattern of a perfect cube: (a − b)³ = a³ − 3a²b + 3ab² − b³ with a = x and b = 2.
f(x)=(x−2)3
Step 3: Identify the multiplicity. The factor (x − 2) appears with exponent 3, so the root x = 2 has multiplicity 3.
Multiplicity of x=2 is 3
Step 4: Verify using derivatives. At a triple root, both the first and second derivatives also equal zero at that point.
f′(x)=3(x−2)2⇒f′(2)=0andf′′(x)=6(x−2)⇒f′′(2)=0
Answer: x = 2 is a triple root of f(x) = x³ − 6x² + 12x − 8 because f(x) = (x − 2)³, giving multiplicity 3.
Another Example
This example differs by having a polynomial with multiple roots of different multiplicities, showing how to distinguish a triple root from a double root within the same polynomial.
Problem: Find all roots and their multiplicities for g(x) = x⁵ − 3x⁴ + 3x³ − x².
Step 1: Factor out the greatest common factor. Every term contains at least x², so factor it out.
g(x)=x2(x3−3x2+3x−1)
Step 2: Factor the cubic. Recognize x³ − 3x² + 3x − 1 as a perfect cube: (x − 1)³.
g(x)=x2(x−1)3
Step 3: Read off the roots and their multiplicities from the factored form.
x=0 with multiplicity 2(double root),x=1 with multiplicity 3(triple root)
Step 4: Check: the sum of multiplicities should equal the degree of the polynomial. 2 + 3 = 5, which matches the degree of g(x).
2+3=5=deg(g)
Answer: The polynomial g(x) has a double root at x = 0 and a triple root at x = 1.
Frequently Asked Questions
What does the graph look like at a triple root?
At a triple root, the graph crosses the x-axis but flattens out as it passes through, creating an inflection point. Unlike a simple root where the curve cuts straight through, the graph of a triple root has a characteristic S-shaped bend that hugs the x-axis momentarily. Both the first and second derivatives equal zero at the triple root, which is why the curve flattens.
What is the difference between a double root and a triple root?
A double root has multiplicity 2, meaning the factor (x − r) appears squared. At a double root, the graph touches the x-axis and bounces back without crossing. A triple root has multiplicity 3, meaning the factor (x − r) appears cubed. At a triple root, the graph crosses the x-axis but flattens at the crossing point. The key visual distinction is that a double root bounces, while a triple root passes through with an inflection.
How do you find a triple root of a polynomial?
Factor the polynomial completely and look for any factor raised to the third power. If (x − r)³ is a factor, then r is a triple root. You can also test a suspected root r by checking that f(r) = 0, f'(r) = 0, and f''(r) = 0, but f'''(r) ≠ 0. If all three conditions hold, r is a triple root.
Triple Root vs. Double Root
| Triple Root | Double Root | |
|---|---|---|
| Multiplicity | 3 | 2 |
| Factor form | (x − r)³ | (x − r)² |
| Graph behavior | Crosses the x-axis with flattening (inflection point) | Touches the x-axis and bounces back |
| Derivative test | f(r) = 0, f'(r) = 0, f''(r) = 0 | f(r) = 0, f'(r) = 0, f''(r) ≠ 0 |
| Minimum polynomial degree | At least degree 3 | At least degree 2 |
Why It Matters
Triple roots appear frequently in algebra and calculus courses when you analyze polynomial behavior and sketch graphs. Recognizing a triple root tells you the graph has an inflection point on the x-axis, which is essential for accurate curve sketching. In applied mathematics and engineering, repeated roots (including triple roots) arise in solving differential equations and indicate special solution structures that differ from the simple-root case.
Common Mistakes
Mistake: Thinking a triple root means the polynomial has three different roots.
Correction: A triple root is a single value that is repeated three times. For example, (x − 4)³ = 0 has one root, x = 4, with multiplicity 3 — not three distinct roots.
Mistake: Assuming the graph bounces off the x-axis at a triple root, like it does at a double root.
Correction: At a triple root, the graph actually crosses the x-axis (because odd multiplicity means a sign change). It flattens at the crossing point but does not bounce. Bouncing behavior occurs only at roots with even multiplicity.
Related Terms
- Root — General term for a solution of a polynomial equation
- Double Root — A root with multiplicity 2 instead of 3
- Multiplicity — The number of times a root is repeated
- Polynomial — The type of expression that has triple roots
- Zero of a Function — An x-value where the function equals zero
- Equation — The statement set to zero to find roots
- Function — The mapping whose zeros include triple roots