By Soo T. Tan
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Extra resources for Applied Mathematics for the Managerial, Life, and Social Sciences (6th Edition)
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 16 CHAPTER 1 FUNDAMENTALS OF ALGEBRA TABLE 7 Factoring Formulas Formula Illustration Difference of two squares a2 Ϫ b2 ϭ 1a ϩ b21a Ϫ b2 x 2 Ϫ 36 ϭ 1x ϩ 621x Ϫ 62 8x 2 Ϫ 2y 2 ϭ 2 14x 2 Ϫ y 2 2 ϭ 2312x2 2 Ϫ y 2 4 ϭ 2(2x ϩ y2(2x Ϫ y2 9 Ϫ a6 ϭ 32 Ϫ 1a3 2 2 ϭ 13 ϩ a3 213 Ϫ a3 2 Perfect square trinomial a2 ϩ 2ab ϩ b2 ϭ 1a ϩ b2 2 a2 Ϫ 2ab ϩ b2 ϭ 1a Ϫ b2 2 x 2 ϩ 8x ϩ 16 ϭ 1x ϩ 42 2 4x 2 Ϫ 4xy ϩ y 2 ϭ 12x2 2 Ϫ 212x21y2 ϩ y 2 ϭ 12x Ϫ y2 2 Sum of two cubes a3 ϩ b3 ϭ 1a ϩ b21a2 Ϫ ab ϩ b2 2 z 3 ϩ 27 ϭ z 3 ϩ 132 3 ϭ 1z ϩ 32 1z 2 Ϫ 3z ϩ 92 Difference of two cubes a3 Ϫ b3 ϭ 1a Ϫ b21a2 ϩ ab ϩ b2 2 8x 3 Ϫ y 6 ϭ 12x2 3 Ϫ 1y 2 2 3 ϭ 12x Ϫ y 2 214x 2 ϩ 2xy 2 ϩ y 4 2 Note Observe that a formula is given for factoring the sum of two cubes, but none is given for factoring the sum of two squares, since x 2 ϩ a 2 is prime over the set of integers.
X 4 Ϫ 16y 2 34. 16u 4√ Ϫ 9√ 3 37. 8m3 ϩ 1 38. 27m3 Ϫ 8 39. 8r 3 Ϫ 27s 3 40. x 3 ϩ 64y 3 2 35. 1a Ϫ 2b 2 2 Ϫ 1a ϩ 2b 2 2 52. a 2 Ϫ b2 ϩ a ϩ b 54. ax 2 Ϫ 1 1 ϩ ab 2 xy ϩ by 2 55. SIMPLE INTEREST The accumulated amount after t years when a deposit of P dollars is made in a bank and earning interest at the rate of r/year is A ϭ P ϩ Prt. Factor the expression on the right-hand side of this equation. 10. 4u√ 1 2u Ϫ √ 2 ϩ 6u √ 1 √ Ϫ 2u 2 2 4 2 36. 2x 1x ϩ y 2 2 Ϫ 8x1x ϩ y 2 2 2 56. M. is Ϫt 3 ϩ 6t 2 ϩ 15t 10 Յ t Յ 4 2 Factor the expression.
But the degree of a term in a polynomial in several variables is obtained by adding the powers of all variables in the term, and the degree of the polynomial is given by the highest degree of all its terms. For example, the polynomial 2x 2y 5 Ϫ 3xy 3 ϩ 8xy 2 Ϫ 3y ϩ 4 is a polynomial in the two variables x and y. It has five terms with degrees 7, 4, 3, 1, and 0, respectively. Accordingly, the degree of the polynomial is 7. Adding and Subtracting Polynomials Constant terms and terms that have the same variable and exponent are called like or similar terms.