Question

What is a perfect square binomial and how do you find the product?

Answer 1

Hint:
A perfect square binomial is a trinomial which when factored gives us the square of a binomial. (Remember that "trinomial" means a "three-term polynomial".) For example, the given trinomial, ${x^2} + 2xy + {y^2}$ is a perfect square binomial because it factors to ${(x + y)^2}$.

Complete step by step:
A perfect square binomial is a trinomial which when factored gives us the square of a binomial.
According to the question we have to define the process of finding the product.
I know that the first term in the original binomial will be the first square root to be discovered, which is $x$. The second term will be the second square root to be found. Looking back at the original quadratic, we need to see whether the sign on the middle term is a "plus" which means that it will have a "plus" sign between the first term and the second term or the middle term has a "minus" sign, that will ensure that it will have a "minus" sign between the first term and the second term.
For instance let’s consider an example, ${x^2}\; + {\text{ }}10x\; + {\text{ }}25$.
When it is further simplified it breaks down into a perfect square $(x + 5){}^2$.

Or, in other words you can say that a squared binomial like $(x + 5){}^2$ can be written as a product of two terms $(x + 5)(x + 5)$.
Which will further yield given result,
$\therefore (x + 5)(x + 5) = {x^2} + 5x + 5x + 5.5$
$ \Rightarrow {x^2} + 10x + 25$

Note:
1) Perfect-square trinomials are of the form: ${a^2}{x^2} + 2axb + {b^2}$
2) Which are further expressed in squared-binomial form as ${(ax \pm b)^2}$.