Question

Expand the given expression ${\left( {1 - 2x} \right)^5}$

Answer 1

Hint – In this question use the direct formula for expansion of ${\left( {1 + x} \right)^n}$ according to binomial expansion which is ${\left( {1 + x} \right)^n} = 1 + nx + \dfrac{{n\left( {n - 1} \right)}}{{2!}}{x^2} + \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{{3!}}{x^3} + ..............$. This will give the answer.

Complete step-by-step solution -
Given equation is
${\left( {1 - 2x} \right)^5}$
Now as we know according to Binomial theorem the expansion of ${\left( {1 + x} \right)^n}$ is
$ \Rightarrow {\left( {1 + x} \right)^n} = 1 + nx + \dfrac{{n\left( {n - 1} \right)}}{{2!}}{x^2} + \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{{3!}}{x^3} + ..............$
So the expansion of ${\left( {1 - 2x} \right)^5}$ according to binomial theorem we have,
$ \Rightarrow {\left( {1 - 2x} \right)^5} = 1 + 5\left( { - 2x} \right) + \dfrac{{5\left( {5 - 1} \right)}}{{2!}}{\left( { - 2x} \right)^2} + \dfrac{{5\left( {5 - 1} \right)\left( {5 - 2} \right)}}{{3!}}{\left( { - 2x} \right)^3} + \dfrac{{5\left( {5 - 1} \right)\left( {5 - 2} \right)\left( {5 - 3} \right)}}{{4!}}{\left( { - 2x} \right)^4} + $
                          $ + \dfrac{{5\left( {5 - 1} \right)\left( {5 - 2} \right)\left( {5 - 3} \right)\left( {5 - 4} \right)}}{{5!}}{\left( { - 2x} \right)^5}$.
Now simplify the above equation we have,
$ \Rightarrow {\left( {1 - 2x} \right)^5} = 1 - 10x + \dfrac{{5\left( 4 \right)}}{{2 \times 1}}\left( {4{x^2}} \right) + \dfrac{{5\left( 4 \right)\left( 3 \right)}}{{3 \times 2 \times 1}}\left( { - 8{x^3}} \right) + \dfrac{{5\left( 4 \right)\left( 3 \right)\left( 2 \right)}}{{4 \times 3 \times 2 \times 1}}\left( {16{x^4}} \right) + \dfrac{{5\left( 4 \right)\left( 3 \right)\left( 2 \right)\left( 1 \right)}}{{5 \times 4 \times 3 \times 2 \times 1}}\left( { - 32{x^5}} \right)$
                                                          \[\left[ {\because n!{\text{ }} = {\text{ }}n\left( {n{\text{ }}-{\text{ }}1} \right)\left( {n{\text{ }}-{\text{ }}2} \right)\left( {n{\text{ }}-{\text{ }}3} \right)\left( {n{\text{ }}-4} \right)\left( {n{\text{ }}-{\text{ }}5} \right).....................} \right]\] $ \Rightarrow {\left( {1 - 2x} \right)^5} = 1 - 10x + 40{x^2} - 80{x^3} + 80{x^4} - 32{x^5}$
So this is the required expansion of${\left( {1 - 2x} \right)^5}$.

Note – Binomial theorem is one which specifies the expansion of any power ${(a + b)^m}$of a binomial $(a + b)$as a certain sum of products ${a^i}{b^j}$ such as ${(a + b)^2} = {a^2} + {b^2} + 2ab$ is also an example a binomial expansion and is derived using this similar concept.