Question

Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it.

$a^2-22a+121$

Short answer

Yes, the given trinomial is a perfect square trinomial.

The factorization of the given trinomial is ${\left(a-11\right)}^{\mathbf{2}}$.

Step-by-step solution

Step 1. Observe the given trinomial a2−22a+121.

The given trinomial is: $a^2-22a+121$

The First, middle, and last terms of the given trinomial are$a^2,-22a$ and 121 respectively.

The first term of the given trinomial can be written as:

$a^2={\left(a\right)}^2$

Therefore, the first term of the given trinomial is a perfect square.

The last term of the given trinomial can be written as:

$121={\left(11\right)}^2$

Therefore, the last term of the given trinomial is a perfect square.

The middle term of the given trinomial can be written as:

$-22a=-2\left(a\right)\left(11\right)$

Therefore, the middle term is twice the product of the square roots of the first term and last term.

Step 2. Determine whether the given trinomial a2−22a+121 is a perfect square trinomial.

As, the first and last terms of the given trinomial are a perfect square and the middle term is twice the product of the square roots of first term and last term.

Therefore, yes, the given trinomial is a perfect square trinomial.

Step 3. Factor the given trinomial a2−22a+121.

It is known that:

$a^2-2ab+b^2=\left(a-b\right)\left(a-b\right)={\left(a-b\right)}^2$

It can be noticed that:

role="math" localid="1647692536357" $$\begin{gathered} a^2-22a+121={\left(a\right)}^2-2\left(a\right)\left(11\right)+{\left(11\right)}^2 \\ =\left(a-11\right)\left(a-11\right)\left(\because a^2-2ab+b^2=\left(a-b\right)\left(a-b\right)={\left(a-b\right)}^2\right) \\ ={\left(a-11\right)}^2 \end{gathered}$$

Therefore, the factorization of the given trinomial is ${\left(a-11\right)}^2$.