Chapter 11.8, Problem 1ST

To determine

To calculate:

Simplification of given expression.

Answer to Problem 1ST

The simplification of ${\left(x-2\right)}^5$ is $x^5-10x^4+40x^3-80x^2+80x-32$ .

Explanation of Solution

Given information:

Expression:

  ${\left(x-2\right)}^5$

Formula used:

Perfect square formula:

  ${\left(a-b\right)}^2=a^2-2ab+b^2$

Where,

  $a=$ first term

  $b=$ second term

Calculation:

Consider expression ${\left(x-2\right)}^5$

Apply the perfect square formula for ${\left(x-2\right)}^5$

  ${\left(a-b\right)}^2=a^2-2ab+b^2$

Where,

  $a=x,b=5$

So, the simplification are as follows,

  $\begin{array}{l}{\left(a-b\right)}^2=a^2-2ab+b^2\\ =x^2-2x.5+5^2\\ =x^2-10+25\end{array}$

The expand summation is,

  $\begin{array}{l}=\frac{5!}{0!\left(5-0!\right)}{x}^5{\left(-2\right)}^0+\frac{5!}{1!\left(5-1!\right)}{x}^4{\left(-2\right)}^1+\mathrm{........}+\frac{5!}{5!\left(5-5!\right)}{x}^0{\left(-2\right)}^5\\ =x^5-10x^4+40x^3-80x^2+80x-32\end{array}$

The answer of given expression is $x^5-10x^4+40x^3-80x^2+80x-32$ .