Chapter 8, Problem 15CT

To determine

To find: the firth term in expansion of the binomial.

Answer to Problem 15CT

Fifth term of expansion of binomial ${\left(2x-10\right)}^6$ is $\underset{\_}{600000x^2}$ .

Explanation of Solution

Given information:

The binomial for which fifth term in expansion needs to be evaluated is as

  ${\left(2x-10\right)}^6$

Formula used:

  $(r+1)$ term of Binomial expansion ${\left(x+y\right)}^n$ is $C{}_r{x}^{n-r}{y}^r$

Where $C{}_r=\frac{n!}{(n-r)!r!}$

Calculation:

Binomial for which fifth term in expansion needs to be evaluated is as

  ${\left(2x-10\right)}^6$

Applying the expansion formula to finding fifth term that is (4+1)th term for ${\left(2x-10\right)}^6$ is $C{}_4{(2x)}^{6-4}{(-10)}^4$

  $=C{}_4{(2x)}^{6-4}{(-10)}^4$

  $=\frac{6!}{(6-4)!4!}{(2x)}^2{(-10)}^4$

  $=\frac{6!}{2!4!}(4x^2)(10000)$

  $=\frac{6\times 5\times 4!}{(2\times 1)4!}(40000x^2)$

  $=15(40000x^2)$

  $=600000x^2$

Hence fifth term of expansion of binomial ${\left(2x-10\right)}^6$ is $\underset{\_}{600000x^2}$ .