Question

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

Answer 1

It is a perfect square. Explanation below.

Explanation:

Perfect squares are of the form `(a + b)^2 = a^2 + 2ab + b^2`. In polynomials of x, the a-term is always x.(`(x+c)^2 = x^2 + 2cx + c^2`)

`x^2 + 8x + 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, `8x` is of the form `2cx`.

The middle term is twice the constant times x, so it is `2xx4xxx = 8x`.

Okay, we found out that the trinomial is of the form `(x+c)^2`, where `x = x and c = 4`.

Let us rewrite it as `x^2 + 8x + 16 = (x+4)^2`. Now we can say it is a perfect square, as it is the square of `(x+4)`.