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.