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Basic of Logarithms
medium
The coefficient of ${x^4}$ in the expansion of the expression ${{3x} \over {(x - 2)(x + 1)}}$ is
A
$-15/16$
B
$15/16$
C
$-16/15$
D
$16/15$
Solution
(b) ${{3x} \over {(x – 2)\,(x + 1)}} = – {{3x} \over 2}{(1 + x)^{ – 1}}{\left( {1 – {x \over 2}} \right)^{ – 1}} = – {3 \over 2}x$
$(1 – x + {x^2} – {x^3} + {x^4} – ..)\,\left( {1 + {x \over 2} + {{\left( {{x \over 2}} \right)}^2} + {{\left( {{x \over 2}} \right)}^3} + …} \right)$
Coefficient of ${x^4} = – {3 \over 2}\left[ { – 1.1 + 1.{1 \over 2} – 1.{1 \over 4} + 1.{1 \over 8}} \right]$
$ = – {3 \over 2}\,\left[ { – 1 + {1 \over 2} – {1 \over 4} + {1 \over 8}} \right] = {{15} \over {16}}$.
Standard 11
Mathematics
