Chapter 1.6, Problem 9QR

Solution

To perform: The indicated operation on an algebraic expression and simplify.

The simplified expression for ${\left(x-1\right)}^3+3{\left(x-1\right)}^2-3\left(x-1\right)$ is $x^3-1$ .

Given data:

The given expression is ${\left(x-1\right)}^3+3{\left(x-1\right)}^2-3\left(x-1\right)$ .

Method/Formula used:

A polynomial having two terms only is called a binomial. The expansions of a binomials ${\left(x+y\right)}^2$ and ${\left(x+y\right)}^3$ are,

  ${\left(x-y\right)}^2=x^2-2xy+y^2$ … (1)

  $\left(x^3-y^3\right)=\left(x-y\right)\left(x^2+xy+y^2\right)$ … (2)

Calculations:

The given expression is ${\left(x-1\right)}^3+3{\left(x-1\right)}^2-3\left(x-1\right)$ . Write it as given below and simplify:

  $\begin{array}{c}E\left(x\right)=\left(x-1\right){}^3+3{\left(x-1\right)}^2-3\left(x-1\right)\leftarrow \text{take}\left(x-1\right)\text{ common factor}\\ =\left(x-1\right)\left\{{\underset{¯}{\left(x-1\right)}}^2+3\left(x-1\right)+3\right\}\leftarrow \text{use}\text{formulae}\left(\text{}\text{1}\right)\\ =\left(x-1\right)\left\{x^2-2x+1+3x-\cancel{3}+\cancel{3}\right\}\leftarrow \text{simplify}\\ =\left(x-1\right)\left\{x^2-2x+3x+1\right\}\leftarrow \text{collect}\text{terms}\text{of}\text{like}\text{powers}\text{of}x\\ =\left(x-1\right)\left(x^2+x+1\right)\leftarrow \text{use}\text{formulae}\left(\text{}\text{2}\right)\\ =x^3-1^3\\ =x^3-1\end{array}$

Therefore, the simplified expression for ${\left(x-1\right)}^3+3{\left(x-1\right)}^2-3\left(x-1\right)$ is $x^3-1$ .