Repeated root

In mathematics, repeated roots (or multiple roots) refer to the situation where a polynomial equation has the same root (solution) more than once. This typically occurs in quadratic, cubic, or higher-degree equations when a factor is squared or raised to a higher power.^[1]

History

Historically, the concept of repeated roots has roots in the development of polynomial equations and their solutions, dating back to ancient civilizations.^[2] The Babylonians and Greeks laid the groundwork for algebraic thought.^[3] The connection between roots and the geometric representation of polynomials was explored during the Renaissance, as mathematicians sought to visualize the solutions on the Cartesian plane.^[2] In the 18th century, the development of calculus by Newton and Leibniz brought a new understanding of polynomial functions and their derivatives. This led to deeper insights into the behavior of polynomials, including the implications of repeated roots on graph behavior, such as tangency to the x-axis.^[4] The formalization of the Fundamental Theorem of Algebra in the 19th century established that every polynomial equation has as many roots as its degree, considering multiplicities. This theorem solidified the importance of repeated roots in the study of polynomials, influencing both theoretical mathematics and practical applications in fields such as engineering and physics.^[5] As mathematics continued to evolve, repeated roots remained a fundamental concept, integral to the understanding of polynomial behavior and solutions.^[6]

Example: Quadratic Equation with Repeated Roots

Consider the quadratic equation:

$(x-2)^{2}=0$

Here, the factor (x−2) appears twice, meaning that x=2 is a repeated root (also called a double root). Expanding the equation:

$(x-2)(x-2)=0$

$x^{2}-4x+4=0$

The discriminant (Δ) of a quadratic equation ax2+bx+c=0 helps to determine whether there are repeated roots:

$\Delta =b^{2}-4ac$

If Δ=0, the equation has one repeated (double) root.

For the equation x2−4x+4=0:

$\Delta =(-4)^{2}-4(1)(4)=16-16=0$

Since the discriminant is zero, this confirms that x=2 is a repeated root.

General Case

In higher-degree polynomials, repeated roots occur when a factor is raised to a power greater than one. For example:

$(x-3)^{3}=0$

In this case, x=3 is a triple root, meaning the solution x=3 is counted three times.

Characteristics of Repeated Roots:

The graph of the polynomial touches the x-axis at the repeated root but does not cross it.
The behavior of the function around repeated roots is "flatter" compared to simple roots.

References

^ Farlow, Stanley J. (1993-01-01). Partial Differential Equations for Scientists and Engineers. Courier Corporation. ISBN 978-0-486-67620-3.
^ ^a ^b Struik, Dirk J. (2012-06-28). A Concise History of Mathematics: Fourth Revised Edition. Courier Corporation. ISBN 978-0-486-13888-6.
^ Stillwell, John (1989). Mathematics and Its History. Springer New York. ISBN 978-0-387-96981-7.
^ Rosen, Kenneth H. (2005). Elementary Number Theory and Its Applications. Pearson/Addison Wesley. ISBN 978-0-321-23707-1.
^ Sinaceur, Hourya (1992). "Cauchy, Sturm et les racines des équations". Revue d'histoire des sciences. 45 (1): 51–67. ISSN 0151-4105.
^ Ritt, Joseph Fels (1950-12-31). Differential Algebra. American Mathematical Soc. ISBN 978-0-8218-4638-4.

[1] Farlow, Stanley J. (1993-01-01). Partial Differential Equations for Scientists and Engineers. Courier Corporation. ISBN 978-0-486-67620-3.

[:0-2] Struik, Dirk J. (2012-06-28). A Concise History of Mathematics: Fourth Revised Edition. Courier Corporation. ISBN 978-0-486-13888-6.

[3] Stillwell, John (1989). Mathematics and Its History. Springer New York. ISBN 978-0-387-96981-7.

[4] Rosen, Kenneth H. (2005). Elementary Number Theory and Its Applications. Pearson/Addison Wesley. ISBN 978-0-321-23707-1.

[5] Sinaceur, Hourya (1992). "Cauchy, Sturm et les racines des équations". Revue d'histoire des sciences. 45 (1): 51–67. ISSN 0151-4105.

[6] Ritt, Joseph Fels (1950-12-31). Differential Algebra. American Mathematical Soc. ISBN 978-0-8218-4638-4.

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