Chapter 1.6, Problem 1QR

Solution

To write: A given expression as a binomial squared.

The required binomial squared of $x^2+2x+1$ is ${\left(x+1\right)}^2$ .

Given data:

The given expression is $x^2+2x+1$ .

Method/Formula used:

A polynomial having two terms only is called a binomial. A polynomial of the form $P={\left(x+y\right)}^2$ is called a binomial squared polynomial. The expansion of a binomial expression ${\left(x+y\right)}^n$ has $n+1$ terms. An expression of a binomial squared ${\left(x+y\right)}^2$ is given by

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

Calculations:

The given The given expression is $x^2+2x+1$ . That can be written as,

  $\left.\begin{array}{c}{x}^2+2x+1={\left(x\right)}^2+2\cdot \left(x\right)\cdot \left(1\right)+\left(1\right)\cdot \left(1\right)\\ ={\left(x\right)}^2+2\cdot \left(x\right)\cdot \left(1\right)+{\left(1\right)}^2\end{array}\right\}$ … (2)

Comparing result (2) with the result (1), the right hand side of the result can be written as:

  $\left.\begin{array}{c}{x}^2+2x+1={\left(x\right)}^2+2\cdot \left(x\right)\cdot \left(1\right)+{\left(1\right)}^2\\ ={\left(x+1\right)}^2\end{array}\right\}$

Therefore, the binomial squared of the expression $x^2+2x+1$ is ${\left(x+1\right)}^2$ .