Chapter 8.7, Problem 9CYU

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 9CYU

The trinomial $y^4+2y^2+1$ is a perfect square and factored trinomial is $y^4+2y^2+1=\left(y^2+1\right)\left(y^2+1\right)$ .

Explanation of Solution

Given information:

The given trinomial is $y^4+2y^2+1$ .

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 $E=y^4+2y^2+1$ that can be written as:

  $E=y^4+2\cdot y^2\cdot +{\left(1\right)}^2$

  $\Rightarrow E={\left(y^2+1\right)}^2$$\because {\left(a+b\right)}^2=a^2+2\cdot a\cdot b+b^2$

Thus, the trinomial $y^4+2y^2+1$ is a perfect square.

The factored trinomial is $y^4+2y^2+1=\left(y^2+1\right)\left(y^2+1\right)$ .

Conclusion:

The trinomial $y^4+2y^2+1$ is a perfect square and factored trinomial is $y^4+2y^2+1=\left(y^2+1\right)\left(y^2+1\right)$ .