By Davenport J.H.

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**Example text**

Of the following two polynomials (this analysis is mostly taken from [Bro71a, Bro71b], but with one change28 : 28 Originally an error, but it makes the point better. 3. GREATEST COMMON DIVISORS 55 −21 instead of +21 for the trailing coefficient of B): A(x) = x8 + x6 − 3x4 − 3x3 + 8x2 + 2x − 5; B(x) = 3x6 + 5x4 − 4x2 − 9x − 21. 2 The first elimination gives A − ( x3 − 92 )B, that is −5 4 127 2 29 x + x − , 9 9 3 and the subsequent eliminations give 50157 2 35847 x − 9x − 25 25 23315940650 93060801700 x+ 1557792607653 173088067517 and, finally, 761030000733847895048691 .

1. WHAT ARE POLYNOMIALS? 1 35 How do we manipulate polynomials? We have defined the abstract, almost Platonic, concept of polynomials as mathematical objects, and polynomial algebra as rules for these objects. What do we mean by the representation of these objects in a computer system? One option would be for a computer algebra system essentially to do nothing, simply recording the computations requested by the user, so that a+b would become simply a + b. However, we would not be very happy with a calculator which computed 1+1 as “1+1”, as we would rather see “2”.

For examples, trigonometry is often encoded by regarding sin θ and cos θ as indeterminates, but subject to sin2 θ + cos2 θ = 1 [Sto77]. Notice what we have not mentioned: division and exponentiation. Definition 21 ((Exact) Division) If a = b ∗ c, then we say that b divides a, and we write b = ac . Note that, for the moment, division is only defined in this context. We note that, if c is not a zero-divisor, b is unique. Definition 22 (Exponentiation) If n is a natural number and a is a polynomial, then we define an inductively by: • a0 = 1; • an+1 = a ∗ an .