Chapter 1.6, Problem 8QR

Solution

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

The simplified expression for ${\left(x-2\right)}^2+3\left(x-2\right)+4$ is $x^2-x+2$ .

Given data:

The given expression is ${\left(x-2\right)}^2+3\left(x-2\right)+4$ .

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)

  ${\left(x+y\right)}^3=x^3+3x^{2}y+3x{y}^2+y^3$ … (2)

Calculations:

The given expression is ${\left(x-2\right)}^2+3\left(x-2\right)+4$ . Write it as given below and simplify:

  $\begin{array}{c}E\left(x\right)=\underset{¯}{{\left(x-2\right)}^2}+3\left(x-2\right)+4\leftarrow \text{use}\text{formula}\left(\text{}1\right)\\ =\underset{¯}{\left(x^2-4x+4\right)}+3\left(x-2\right)+4\leftarrow \text{simplify}\\ =x^2-4x+4+3x-6+4\\ =x^2+\left(-4+3\right)x+\left(-6+4+4\right)\leftarrow \text{collect}\text{terms}\text{of}\text{like}\text{powers}\text{of}x\\ =x^2+\left(-1\right)x+\left(2\right)\\ =x^2-x+2\end{array}$

Therefore, the simplified expression for ${\left(x-2\right)}^2+3\left(x-2\right)+4$ is $x^2-x+2$ .