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Names of Polynomials


Date: 05/22/99 at 15:59:52
From: Rohan
Subject: Why is an equation having only two roots, one of which is 
raised to 2, called a "quadratic equation"?

QUAD means four. But why is an equation having ONLY TWO ROOTS, one of 
which is raised to 2, called a "QUADRATIC EQUATION"?

For example: - ax^2 + bx + c


Date: 05/22/99 at 20:40:32
From: Doctor Peterson
Subject: Re: Why is an equation having only two roots, one of which is 
raised to 2, called a "quadratic equation"?

Hi, Rohan.

People often wonder about the word "quadratic," because they know that 
"quad" usually means "four," yet quadratic equations involve the 
second power, not the fourth. But there's another dimension to the 
word.

Although in Latin the prefix "quadri" means four, the word "quadrus" 
means a square (because it has four sides) and "quadratus" means 
"squared." We get several other words from this: "quadrille," meaning 
a square dance; "quadrature," meaning constructing a square of a 
certain area; and even "square" (through French).

Quadratic equations originally came up in connection with geometric 
problems involving squares, and of course the second power is also 
called a "square," which accounts for the name. The third-degree 
equation is similarly called a "cubic," based on the shape of a third 
power. Then when higher-degree equations began to be studied, the 
names for them were formed differently, based on degree rather than 
shape (since the Romans had no words for higher-dimensional shapes), 
giving us the quartic, quintic, and so on. In fact, quartic came 
along later; originally a fourth degree equation was called 
"biquadratic," meaning "doubly squared," which mixes the two concepts 
and is doubly confusing.

So here's a table of names for polynomials and their sources:

    degree   name        shape    dimension
    ------   ---------   ------   ---------
       1     linear      line        (1)
       2     quadratic   square      (2)
       3     cubic       cube        (3)
       4     quartic      -           4
       5     quintic      -           5

You may be interested in a discussion of this question in the archives 
of the math-history discussion group:

http://mathforum.org/kb/message.jspa?messageID=1376346   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
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