812 CHAPTER 11 Systems of Equations and Inequalities Consider the problem of adding two rational expressions: + − x x 3 4 and 2 3 The sum is ( ) ( ) ( )( ) + + − = − + + + − = − + − x x x x x x x x x 3 4 2 3 3 3 2 4 4 3 5 1 12 2 OBJECTIVES 1 Decompose P Q where Q Has Only Nonrepeated Linear Factors (p. 814) 2 Decompose P Q where Q Has Repeated Linear Factors (p. 815) 3 Decompose P Q where Q Has a Nonrepeated Irreducible Quadratic Factor (p. 817) 4 Decompose P Q where Q Has a Repeated Irreducible Quadratic Factor (p. 818) 11.5 Partial Fraction Decomposition • Identity (Section A.6, p. A44) • Properties of Rational Functions (Section 4.5, pp. 236–244) • Reducing a Rational Expression to Lowest Terms (Section A.5, pp. A35–A36) • Complex Zeros; Fundamental Theorem of Algebra (Section 4.4, pp. 229–234) PREPARING FOR THIS SECTION Before getting started, review the following: Now Work the ‘Are You Prepared?’ problems on page 819. 94. Challenge Problem If = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A a b c d and = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ B 1 1 1 1 , find a, b, c, d so that AB BA. = Explaining Concepts 96. Create a situation different from any found in the text that can be represented by a matrix. 97. Explain why the number of columns in matrix A must equal the number of rows in matrix B to find the product AB. 98. If a, b, and c 0 ≠ are real numbers with ac bc, = then a b. = Does this same property hold for matrices? In other words, if A, B, and C are matrices and AC BC, = must A B? = 99. What is the solution of the system of equations AX 0 = if A 1− exists? Discuss the solution of AX 0 = if A 1− does not exist. Retain Your Knowledge Problems 100–109 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 100. Find a polynomial with minimum degree and leading coefficient 1 that has zeros x 3 = (multiplicity 2), x 0 = (multiplicity 3), and x 2 = − (multiplicity 1). 101. For v i j 2 = − − and w i j 2 , = + find the dot product v w⋅ and the angle between v and w. 102. Solve: x x x x 5 2 2 + = − 103. Write ( ) − u cos csc 1 as an algebraic expression in u. 104. Express e8 i /3 π⋅ in rectangular form. 105. Add: x x x 1 3 4 3 + − + + 106. Find the domain of f x x x 10 2 3 . ( ) = − + 107. Factor completely: x x x x 3 12 108 432 4 3 2 + − − 108. Find the area of the region enclosed by the graphs of y x 4 , 2 = − y x 2, = − and y x 2. = − − 109. If f x x x 25 4 2 ( ) = − and g x x x 2 5 sec , 0 2 , π ( ) = < < show that f g x x 5 sin . ( )( ) = 95. Challenge Problem If = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A a b b a , find a and b so that A A 0. 2 + =
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