36 Chapter 2Algebraic functions
Therefore
x
41 − 15 x
21 + 141 = 1 (x
21 − 1 1)(x
21 − 1 4)
Both the quadratic factors have the formx
21 − 1 a
21 = 1 (x 1 + 1 a)(x 1 − 1 a)discussed in
case (iii) above:
x
21 − 111 = 1 (x 1 + 1 1)(x 1 − 1 1) and x
21 − 141 = 1 (x 1 + 1 2)(x 1 − 1 2)
Therefore,
x
41 − 15 x
21 + 141 = 1 (x 1 + 1 1)(x 1 − 1 1)(x 1 + 1 2)(x 1 − 1 2)
0 Exercises 9–16
The expansion
(x 1 + 1 a)(x 1 + 1 b) 1 = 1 x
21 + 1 (a 1 + 1 b)x 1 + 1 ab (2.4)
used in Examples 2.6, has geometric interpretation as the area of a rectangle of sides
(x 1 + 1 a)and(x 1 + 1 b), as illustrated in Figure 2.3.
3Other important general forms are
(a 1 + 1 b)
21 = 1 a
21 + 12 ab 1 + 1 b
2square of side (a 1 + 1 b)
(a 1 − 1 b)
21 = 1 a
21 − 12 ab 1 + 1 b
2square of side |a 1 − 1 b| (2.5)
(a 1 + 1 b)(a 1 − 1 b) 1 = 1 a
21 − 1 b
2difference of squares
The first two equations of (2.5) can be combined by using the symbol ±, meaning ‘plus
or minus’:
(a 1 ± 1 b)
21 = 1 a
21 ± 12 ab 1 + 1 b
2(2.6)
in which either the upper symbol is used on bothsides of the equation or the lower
symbol is used on both sides. Sometimes the symbol 3 is used in a similar way; for
example,a 131 b 1 = 1 ±crepresents the pair of equationsa 1 − 1 b 1 = 1 +canda 1 + 1 b 1 = 1 −c.
Factorization can be used to simplify algebraic fractions. For example, in
xy x
xxy
2
46
23Euclid, ‘The Elements’, Book II, Propositions 4 and 7 are the geometric equivalents of the first two equations
(2.5) for the squares of (a 1 + 1 b) and (a 1 − 1 b).
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................................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ......................................................................bxab
x
2ax
b
x
xa
Figure 2.3