Download An elementary treatise on elliptic functions by Arthur Cayley PDF

By Arthur Cayley

This quantity is made out of electronic photographs from the Cornell collage Library old arithmetic Monographs assortment.

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2͑3 Ϫ Ί7 ͒ ͑3͒2 Ϫ ͑Ί7 ͒2 Find products. In denominator, ͑a ϩ b͒͑a Ϫ b͒ ϭ a 2 Ϫ ab ϩ ab Ϫ b 2 ϭ a 2 Ϫ b 2. 2͑3 Ϫ Ί7 ͒ ϭ 3 Ϫ Ί7 2 Simplify and divide out common factors. Now try Exercise 93. In calculus, sometimes it is necessary to rationalize the numerator of an expression. STUDY TIP Example 13 Rationalizing a Numerator Rationalize the numerator of Ί5 Ϫ Ί7 2 . Solution Ί5 Ϫ Ί7 2 ϭ Ί5 Ϫ Ί7 Ί5 ϩ Ί7 и Ί5 ϩ Ί7 2 ͑Ί5 ͒2 Ϫ ͑Ί7͒ ϭ 2͑Ί5 ϩ Ί7 ͒ 2 ϭ Ϫ2 Ϫ1 ϭ 2͑Ί5 ϩ Ί7͒ Ί5 ϩ Ί7 Multiply numerator and denominator by conjugate of numerator.

X 2 ϩ x Ϫ 2 102. x 2 ϩ 5x ϩ 6 103. s 2 Ϫ 5s ϩ 6 104. t 2 Ϫ t Ϫ 6 105. 20 Ϫ y Ϫ y 2 106. 24 ϩ 5z Ϫ z 2 107. 3x 2 Ϫ 5x ϩ 2 108. 3x2 ϩ 13x Ϫ 10 109. 2x 2 ϪxϪ1 ϩ 4x Ϫ 128. 16 Ϫ 6x Ϫ x2 130. 7y 2 ϩ 15y Ϫ 2y3 2x 3 110. 2x2 1 2 8x Ϫ 1 96 x Ϫ 132. 13x ϩ 6 ϩ 5x 2 1 16 134. 135. 3x 3 ϩ x 2 ϩ 15x ϩ 5 137. 3u Ϫ 2u2 1 2 81 x 2 ϩ 9x Ϫ 8 136. 5 Ϫ x ϩ 5x 2 Ϫ x 3 ϩ6Ϫu 3 139. 2x3 ϩ x2 Ϫ 8x Ϫ 4 83. x 2 Ϫ 4x ϩ 4 4 2x 2 138. x 4 Ϫ 4x 3 ϩ x 2 Ϫ 4x In Exercises 83–90, factor the perfect square trinomial. 89. 4x2 Ϫ 3x ϩ 9 129.

Solution When multiplying two polynomials, be sure to multiply each term of one polynomial by each term of the other. A vertical format is helpful. 4x2 ϩ x Ϫ 2 Write in standard form. Ϫx ϩ 3x ϩ 5 Write in standard form. 20x2 ϩ 5x Ϫ 10 5͑4x 2 ϩ x Ϫ 2͒ 2 ϫ 12x3 ϩ 3x2 Ϫ 6x Ϫ4x4 Ϫ 3x͑4x 2 ϩ x Ϫ 2͒ x3 ϩ 2x2 Ϫx 2͑4x 2 ϩ x Ϫ 2͒ Ϫ4x4 ϩ 11x3 ϩ 25x2 Ϫ x Ϫ 10 Combine like terms. Now try Exercise 59. Special Products Special Products Let u and v be real numbers, variables, or algebraic expressions. Special Product Example Sum and Difference of Same Terms ͑u ϩ v͒͑u Ϫ v͒ ϭ u 2 Ϫ v 2 Square of a Binomial ͑x ϩ 4͒͑x Ϫ 4͒ ϭ x 2 Ϫ 42 ϭ x2 Ϫ 16 ͑u ϩ v͒ 2 ϭ u 2 ϩ 2uv ϩ v 2 ͑u Ϫ v͒ 2 ϭ u 2 Ϫ 2uv ϩ v 2 ͑x ϩ 3͒ 2 ϭ x 2 ϩ 2͑x͒͑3͒ ϩ 32 ϭ x2 ϩ 6x ϩ 9 ͑3x Ϫ 2͒2 ϭ ͑3x͒2 Ϫ 2͑3x͒͑2͒ ϩ 22 ϭ 9x2 Ϫ 12x ϩ 4 Cube of a Binomial ͑u ϩ v͒3 ϭ u 3 ϩ 3u 2v ϩ 3uv 2 ϩ v 3 ͑u Ϫ v͒3 ϭ u 3 Ϫ 3u 2v ϩ 3uv 2 Ϫ v 3 ͑x ϩ 2͒3 ϭ x 3 ϩ 3x 2͑2͒ ϩ 3x͑22͒ ϩ 23 ϭ x3 ϩ 6x2 ϩ 12x ϩ 8 ͑x Ϫ 1͒3 ϭ x 3 Ϫ 3x 2͑1͒ ϩ 3x͑12͒ Ϫ 13 ϭ x3 Ϫ 3x2 ϩ 3x Ϫ 1 Example 5 The Product of Two Trinomials Find the product of x ϩ y Ϫ 2 and x ϩ y ϩ 2.

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