Chapter 8, Problem 54CR

To determine

To find: The expand and simplification of the given expression.

Answer to Problem 54CR

The simplification is $y^3-9y^2+27y-27$ .

Explanation of Solution

Given:

The given expression is,

  ${\left(y-3\right)}^3$

Calculation:

The binomial expansion is written as,

  ${\left(x+y\right)}^n=x^n+n{x}^{n-1}y+\mathrm{...}+C{}_r{x}^{n-r}{y}^r+\mathrm{...}+nx{y}^{n-1}+y^n$

The formula is written as,

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

The binomial coefficients are calculated as,

  $\begin{array}{c}C{}_0=1\\ C{}_1=3\\ C{}_2=3\\ C{}_3=1\end{array}$

The expansion for the given expression is written as,

  $\begin{array}{c}{\left(y-3\right)}^3=\left(1\right){\left(y\right)}^3-\left(3\right)y^{3-1}{\left(3\right)}^1+\left(3\right)y^{3-2}{\left(3\right)}^2-\left(1\right)y^{3-3}{\left(3\right)}^3\\ =y^3-9y^2+27y-27\end{array}$

Therefore, the simplification is $y^3-9y^2+27y-27$ .