Chapter 8.7, Problem 10CYU

To determine

To find: That the given polynomial is a perfect square trinomial, write yes or no. If yes then factor it.

Answer to Problem 10CYU

The trinomial $25x^2{y}^2-20xy+4y^2$ is not a perfect square.

Explanation of Solution

Given information:

The given trinomial is $25x^2{y}^2-20xy+4y^2$ .

Concept used:

For polynomial $a{x}^2+bx+c$ is perfect square trinomial if it can be written as $a{x}^2+bx+c={\left(mx+n\right)}^2=m{x}^2+2\cdot mx.n+n^2$ , i.e. we can find two equal factors of $a\times c$ whose sum equals b .

Direct method for $a{x}^2+bx+c$ check whether $b^2=4ac$ , if yes then $a{x}^2+bx+c$ will be a perfect square. Also, if $b^2<4ac$ the trinomial cannot be factorized.

Calculations:

The given trinomial E (say) is $E=25x^2{y}^2-20xy+4y^2$ .

  $E=\left(5xy\right)-2\cdot 5xy\cdot 2\left(\right)+{\left(2y\right)}^2$

Since, the trinomial $E=25x^2{y}^2-20xy+4y^2$ cannot be written as

  $a{x}^2+bx+c={\left(mx+n\right)}^2=m{x}^2+2\cdot mx.n+n^2$ . So, it is not a perfect square.

Conclusion:

The trinomial $25x^2{y}^2-20xy+4y^2$ is not a perfect square.