By Todhunter, I. (Isaac)
The beneficial reception which has been granted to my historical past of the Calculus of adaptations through the 19th Century has inspired me to adopt one other paintings of an analogous variety. the topic to which I now invite realization has excessive claims to attention as a result of the sophisticated difficulties which it comprises, the precious contributions to research which it has produced, its very important sensible functions, and the eminence of these who've cultivated it.
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Extra info for A history of the mathematical theory of probability : from the time of Pascal to that of Laplace
9 = (x + 3) (x - 3) The general rule, which holds whenever we squares, may be stated as follows. I are dealing with a difference of two a2 - b2 = (a + b) (a - b) Dlllerence of Two Squares EXAMPLE 5 Factor. - - 25 (h)'2 - (5)2 = = (h + 5)(h - 5 ) = (3r we may use the formula for the difference of two squares with b 5. - ANSWERS (a) (x 7)(x - 7) (b) - ( )2 4t, resulting in 9r2 - 16t2 PROGRESS CHECK Factor. - (4x + 3)(4x - 3) - y2 (c) (5x + y)(5x - y) The fonnulas for a sum of two cubes and a difference of two cubes can be verified by multiplying the factors on the right-hand sides of the following equa tions.
5 In Exercises 30-33 perfonn the indicated operations and simplify. - I) 9(x + y) 4 - x2 x - 2 30. 3(14(y x2 - y2) _ ---:::;;;:- 3 1 . 2Y2 3Y b a2 - 4b2 32· aa ++ 2b · a2 b2 33 · x2 - 2x - 3 x2 - 4x + 3 i? - x 3x3 - 3x2 In Exercises 34-37 find the LCD. 3 34. - I 2 2x2 ' x2 - 4 ' x - 2 -3 5 35. " 5(x 1 )2 x-2 y- 1 3x 37· x2(y + I)' 2xy - 2x' 4y2 + Sy + 4 ber system that justifies the statement. All variables represent real numbers. 8. 9. IO. 11. 2 In Exercises 1 2- 14 sketch the given set of numbers on - z + a real number line.
Integers (p. 2) - bi is the additive inverse of the comnumber a + bi. degree of a polynomial (p. 1 7) 1 2) ( �) (3 - 2i) - 2 + i + Oi is the additive identity and I + Oi is set ( p . I ) - 3i 30. the multi plicative identity for the set of complex num TERMS AND SYM BOLS - 2 - V-16 a + bi. numbers. bers. 21. 2 + 3i + (3 - 2i) holds for the set of complex numbers . Prove that 0 3 - V-49 (2y + I ) - (2x - l )i = -8 + 3i plex Prove thal the commutative law of multip lication 17. 29. the set of complex numbers .