Chapter 1.6, Problem 6QR

Solution

To write: A given expression as a binomial squared.

The required binomial squared of $4x^2-20x+25$ is ${\left(2x-5\right)}^2$ .

Given data:

The given expression is $4x^2-20x+25$ .

Method/Formula used:

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

  $\left.\begin{array}{l}{\left(a+b\right)}^2={\left(a\right)}^2+2\cdot \left(a\right)\cdot \left(b\right)+{\left(b\right)}^2\\ {\left(a-b\right)}^2={\left(a\right)}^2-2\cdot \left(a\right)\cdot \left(b\right)+{\left(b\right)}^2\end{array}\right\}$ … (1)

Calculations:

The given expression is $4x^2-20x+25$ . That can be written as,

  $\left.\begin{array}{c}4x^2-20x+25={\left(2x\right)}^2-2\cdot \left(2x\right)\cdot \left(5\right)+\left(5\right)\cdot \left(5\right)\\ ={\left(2x\right)}^2-2\cdot \left(2x\right)\cdot \left(5\right)+{\left(5\right)}^2\end{array}\right\}$ … (2)

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

  $\left.\begin{array}{c}4x^2-20x+25={\left(2x\right)}^2-2\cdot \left(2x\right)\cdot \left(5\right)+{\left(5\right)}^2\\ ={\left(2x-5\right)}^2\end{array}\right\}$

Therefore, the binomial squared of the expression $4x^2-20x+25$ is ${\left(2x-5\right)}^2$ .