Chapter 8.7, Problem 7CYU

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 7CYU

The trinomial $25x^2+60x+36$ is a perfect square and the factored trinomial is $25x^2+60x+36=\left(5x+6\right)\left(5x+6\right)$ .

Explanation of Solution

Given information:

The given trinomial is $25x^2+60x+36$ .

Concept used:

For polynomial $a{x}^2+bx-c$ , find two factors of $a\times c$ such that their difference is b.

Calculations:

The given trinomial E (say) is $E=25x^2+60x+36$ .

We try to find two factors of $25\times 36$ such that their sum is 60 as follows:

  $25\times 36=5\times 5\times 2\times 2\times 3\times 3$

  $\Rightarrow 25\times 36=\left(5\times 2\times 3\right)\times \left(5\times 2\times 3\right)$

  $\Rightarrow 25\times 36=\left(30\right)\times \left(30\right)$

Such that $30+30=60$ .

(Another, 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: Here ${60}^2=4\times 25\times 36=3600$ )

That shows that the polynomial $E=25x^2+60x+36$ is a perfect square trinomial.

Thus, the trinomial $E$ can be written as $E={\left(5x\right)}^2+2\cdot 5x\cdot 6+{\left(6\right)}^2$

  $\Rightarrow E={\left(5x+6\right)}^2=\left(5x+6\right)\left(5x+6\right)$

Thus, the factored trinomial is $E=25x^2+60x+36=\left(5x+6\right)\left(5x+6\right)$ .

Conclusion:

The trinomial $25x^2+60x+36$ is a perfect square and so can be factorized, and the factored trinomial is $25x^2+60x+36=\left(5x+6\right)\left(5x+6\right)$ .