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$(1 - x) (1 - 2x) (1 - 2^2. x) (1 - 2^3. x) …. (1 - 2^{15}. x) $ ના ગુણાકારમાં $x^{15} $ નો સહગુણક મેળવો.
$2^{105} - 2^{121}$
$2^{121} - 2^{105}$
$2^{120} - 2^{104}$
આપેલ પૈકી એકપણ નહિ.
Solution
ગુણાકાર = $( – 1)\,( – 2)\,( – {2^2})\,…\,( – {2^{15}})\,(x\, – \,1)\,\left( {x\,\, – \,\,\frac{1}{2}} \right)\,\left( {x\,\, – \,\,\frac{1}{{{2^2}}}} \right)\,\left( {x\,\, – \,\,\frac{1}{{{2^3}}}} \right)\,…\,\left( {x\,\, – \,\,\frac{1}{{{2^{15}}}}} \right)$
$=\, – \,{2^{1 + 2 + 3\,…\, + \,15}}\,(x\, – \,1)\,\left( {x\,\, – \,\,\frac{1}{2}} \right)\,\left( {x\,\, – \,\,\frac{1}{{{2^2}}}} \right)\,….\,\left( {x\,\, – \,\,\frac{1}{{{2^{15}}}}} \right)$
$x^{15}$ નો સહગુણક $\, = \,\,{2^{1\, + \,2\, + \,3\, + \,…..\,15}}\,\left( { – 1\,\, – \,\,\frac{1}{2}\,\, – \,\,\frac{1}{{{2^2}}}\, – \,….\, – \,\frac{1}{{{2^{15}}}}} \right)$
$x^{15}$ નો સહગુણક $\, =2^{105} – 2^{121}$
