Q11. - Algebra 1

Question

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

$x^2+20x+100$

Step-by-Step Solution

Answer

Yes, the given trinomial is a perfect square trinomial.

The factorization of the given trinomial is ${\left(x+10\right)}^2$.

Step 1. Observe the given trinomial x 2 + 20 x + 100 .

The given trinomial is: $x^2+20x+100$

The first, middle, and last terms of the given trinomial are $x^2,20x$ and 100 respectively.

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

 $x^2={\left(x\right)}^2$

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

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

 $100={\left(10\right)}^2$

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

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

  $20x=2\left(x\right)\left(10\right)$

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 x 2 + 20 x + 100 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 the first term and last term.

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

Step 3. Factor the given trinomial x 2 + 20 x + 100 .

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:

 $$\begin{gathered} x^2+20x+100={\left(x\right)}^2+2\left(x\right)\left(10\right)+{\left(10\right)}^2 \\ =\left(x+10\right)\left(x+10\right)\left(\because a^2+2ab+b^2=\left(a+b\right)\left(a+b\right)={\left(a+b\right)}^2\right) \\ ={\left(x+10\right)}^2 \end{gathered}$$

Therefore, the factorization of the given trinomial is ${\left(x+10\right)}^2$.