SECTION A.4 Synthetic Division A31 69. − + + x x x 2 3 1 4 3 divided by + + x x 2 1 2 70. − + − x x x 3 2 4 3 divided by + + x x 3 1 2 71. − + − x x 4 4 3 2 divided by −x 1 72. − − − x x 3 2 1 4 divided by −x 1 73. − + x x 1 2 4 divided by + + x x 1 2 74. − + x x 1 2 4 divided by − + x x 1 2 75. − x a 3 3 divided by −x a 76. − x a 5 5 divided by −x a In Problems 77–124, factor completely each polynomial. If the polynomial cannot be factored, say it is prime. 77. − x 36 2 78. − x 9 2 79. − x 2 8 2 80. − x 3 27 2 81. + + x x 11 10 2 82. + + x x5 4 2 83. − + x x 10 21 2 84. − + x x6 8 2 85. − + x x 4 8 32 2 86. − + x x 3 12 15 2 87. + + x x4 16 2 88. + + x x 12 36 2 89. + −x x 15 2 2 90. + −x x 14 6 2 91. − − x x 3 12 36 2 92. + − x x x 8 20 3 2 93. + + y y y 11 30 4 3 2 94. − − y y y 3 18 48 3 2 95. + + x x 4 12 9 2 96. − + x x 9 12 4 2 97. + + x x 6 8 2 2 98. + − x x 8 6 2 2 99. − x 81 4 100. − x 1 4 101. − + x x2 1 6 3 102. + + x x2 1 6 3 103. − x x 7 5 104. − x x 8 5 105. + + x x 16 24 9 2 106. − + x x 9 24 16 2 107. + −x x 5 16 16 2 108. + −x x 5 11 16 2 109. − + y y 4 16 15 2 110. + − y y 9 9 4 2 111. − − x x 1 8 9 2 4 112. − − x x 4 14 8 2 4 113. ( ) ( ) + − + x x x 3 6 3 114. ( ) ( ) − + − x x x 5 3 7 3 7 115. ( ) ( ) + − + x x 2 5 2 2 116. ( ) ( ) − − − x x 1 2 1 2 117. ( ) − − x3 2 27 3 118. ( ) + − x5 1 1 3 119. ( ) ( ) + + − + x x x 3 10 25 4 5 2 120. ( ) ( ) − + + − x x x 7 6 9 5 3 2 121. + − − x x x 2 2 3 2 122. − − + x x x 3 3 3 2 123. − + − x x x 1 4 3 124. + + + x x x 1 4 3 In Problems 125–130, determine the number that should be added to complete the square of each expression. Then factor each expression. 125. + x x 10 2 126. + p p 14 2 127. − y y6 2 128. − x x4 2 129. − x x 1 2 2 130. + x x 1 3 2 Applications and Extensions In Problems 131–140, expressions that occur in calculus are given. Factor completely each expression. 131. ( ) ( ) ( ) + + + ⋅ + ⋅ x x x 2 3 4 2 3 2 3 4 3 2 132. ( ) ( ) ( ) + + − ⋅ + ⋅ x x x 5 2 1 5 6 2 2 1 2 2 133. ( ) + + ⋅ x x x 2 2 5 2 2 134. ( ) − + ⋅ x x x 3 8 3 8 2 3 135. ( )( ) ( ) ( ) + − + + ⋅ − x x x x 2 3 2 3 3 2 3 2 2 136. ( ) ( ) ( ) ( ) + − + + ⋅ − x x x x 4 5 1 5 2 1 3 2 4 137. ( ) ( ) − + ⋅ − ⋅ x x x 4 3 2 4 3 4 2 138. ( ) ( ) + + ⋅ + ⋅ x x x x 3 3 4 2 3 4 3 2 2 3 139. ( ) ( ) ( ) ( ) − ⋅ + + − ⋅ + ⋅ x x x x 2 3 5 3 2 1 3 5 3 2 1 2 3 2 2 140. ( ) ( ) ( ) ( ) + ⋅ + + + ⋅ + ⋅ x x x x 3 4 5 4 5 1 4 5 2 5 1 5 2 2 3 In Problems 141–144, factor each expression by factoring out the like term(s) with the smallest exponent. For example, ( ) + = + − − − x x x x 1 2 3 3 141. + − − x x 4 2 142. − + − − − x x x 2 6 8 5 4 3 143. ( ) ( ) + + − − − x x x x 1 1 2 1 144. ( ) ( ) + + + − − x x x x 3 4 3 1 2 2 145. Show that + x 4 2 is prime. 146. Show that + + x x 1 2 is prime. Explaining Concepts 147. Make up a polynomial that factors into a perfect square. 148. Explain to a fellow student what you look for first when presented with a factoring problem. What do you do next? 1 Divide Polynomials Using Synthetic Division To find the quotient as well as the remainder when a polynomial of degree 1 or higher is divided by −x c, a shortened version of long division, called synthetic division , makes the task simpler. A.4 Synthetic Division OBJECTIVE 1 Divide Polynomials Using Synthetic Division (p. A31)
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