# How do you know if  x^2 + 8x + 16 is a perfect square trinomial and how do you factor it?

May 2, 2018

It is a perfect square. Explanation below.

#### Explanation:

Perfect squares are of the form ${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$. In polynomials of x, the a-term is always x.(${\left(x + c\right)}^{2} = {x}^{2} + 2 c x + {c}^{2}$)

${x}^{2} + 8 x + 16$ is the given trinomial. Notice that the first term and the constant are both perfect squares: ${x}^{2}$ is the square of x and 16 is the square of 4.

So we find that the first and last terms correspond to our expansion. Now we must check if the middle term, $8 x$ is of the form $2 c x$.

The middle term is twice the constant times x, so it is $2 \times 4 \times x = 8 x$.

Okay, we found out that the trinomial is of the form ${\left(x + c\right)}^{2}$, where $x = x \mathmr{and} c = 4$.

Let us rewrite it as ${x}^{2} + 8 x + 16 = {\left(x + 4\right)}^{2}$. Now we can say it is a perfect square, as it is the square of $\left(x + 4\right)$.