sat suite question viewer

Advanced Math / Nonlinear functions Difficulty: Hard

The function $g$ is defined by $g (x) = x (x - 2) {(x + 6)}^{2}$ . The value of $g left parenthesis 7 minus w right parenthesis$ is $0$ , where $w$ is a constant. What is the sum of all possible values of $w$ ?

Back question 178 of 229 Next

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229

Explanation

The correct answer is $25$ . The value of $g left parenthesis 7 minus w right parenthesis$ is the value of $g (x)$ when $x = 7 - w$ , where $w$ is a constant. Substituting $7 minus w$ for $x$ in the given equation yields $g (7 - w) = (7 - w) (7 - w - 2) {(7 - w + 6)}^{2}$ , which is equivalent to $g (7 - w) = (7 - w) (5 - w) {(13 - w)}^{2}$ . It’s given that the value of $g left parenthesis 7 minus w right parenthesis$ is $0$ . Substituting $0$ for $g left parenthesis 7 minus w right parenthesis$ in the equation $g (7 - w) = (7 - w) (5 - w) {(13 - w)}^{2}$ yields $0 = (7 - w) (5 - w) {(13 - w)}^{2}$ . Since the product of the three factors on the right-hand side of this equation is equal to $0$ , at least one of these three factors must be equal to $0$ . Therefore, the possible values of $w$ can be found by setting each factor equal to $0$ . Setting the first factor equal to $0$ yields $7 - w = 0$ . Adding $w$ to both sides of this equation yields $7 equals w$ . Therefore, $7$ is one possible value of $w$ . Setting the second factor equal to $0$ yields $5 - w = 0$ . Adding $w$ to both sides of this equation yields $5 equals w$ . Therefore, $5$ is a second possible value of $w$ . Setting the third factor equal to $0$ yields ${(13 - w)}^{2} = 0$ . Taking the square root of both sides of this equation yields $13 - w = 0$ . Adding $w$ to both sides of this equation yields $13 equals w$ . Therefore, $13$ is a third possible value of $w$ . Adding the three possible values of $w$ yields $7 + 5 + 13$ , or $25$ . Therefore, the sum of all possible values of $w$ is $25$ .