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MathidiaMathematics Academy
Polynomial roots
Lesson 13 of 75
Standard practice

Repeated roots

Objective. Connect common factors of a polynomial and its derivative.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

A repeated root of a polynomial p(x) is a root shared by p(x) and its derivative p'(x), since near a repeated root the graph touches rather than crosses the x-axis. To find repeated roots, compute the greatest common factor of p(x) and p'(x); any common root of both is a repeated root of p(x). For example, p(x) = x³ − 3x + 2 has p'(x) = 3x² − 3, and checking x=1 gives p(1)=0 and p'(1)=0, confirming x=1 is a repeated root of p.

Worked example

Show that x³ − 5x² + 8x − 4 has a repeated root, and find its value.

5 … 8
Illustrated card showing 5 … 8
  1. 1Let p(x) = x³ − 5x² + 8x − 4, so p'(x) = 3x² − 10x + 8
  2. 2Solve p'(x)=0: discriminant = 100 − 96 = 4, roots x = (10±2)/6 = 2 or 4/3
  3. 3Test x=2 in p: 8 − 20 + 16 − 4 = 0, so x=2 is a common root
  4. 4Since p(2)=0 and p'(2)=0, x=2 is a repeated root

Answer: x = 2 is the repeated root

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Common mistake. Students assume any root of p'(x) = 0 is automatically a repeated root of p(x), forgetting it must also satisfy p(x) = 0. A turning point of p is not a root of p unless p also equals zero there.

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