Chapter 6, Problem 2TP

Solution

To find the fourth term in the expansion of ${\left(x-3\right)}^6$.

The fourth term in the expansion of the binomial is $-540x^3$.

Given information:

The binomial ${\left(x-3\right)}^6$.

Formula used:

The $r\text{th}$ term in the binomial expansion ${\left(a+b\right)}^n$ is given by $C{}_{r-1}\cdot a^{n-r+1}\cdot b^{r-1}$.

Calculation:

Compare ${\left(x-3\right)}^6$ with ${\left(a+b\right)}^n$ to get $a=x,b=-3$ and $n=6$.

Now, substitute $a=x,b=-3$ , $n=6$ and $r=4$ in the formula $C{}_{r-1}\cdot a^{n-r+1}\cdot b^{r-1}$ to find the 4^th term of the binomial expansion.

  $\begin{array}{c}C{}_{4-1}\cdot x^{6-4+1}\cdot {\left(-3\right)}^{4-1}=C{}_3\cdot x^3\cdot {\left(-3\right)}^3\\ =\frac{6!}{\left(6-3\right)!\cdot 3!}\cdot x^3\left(-27\right)\\ =\frac{6!}{3!\cdot 3!}\cdot x^3\left(-27\right)\\ =\frac{6\cdot 5\cdot 4}{3!}\cdot x^3\left(-27\right)\\ =-540x^3\end{array}$

Conclusion:

The fourth term in the expansion of the binomial is $-540x^3$.