Perfect Square Polynomials

Perfect square polynomials are a special kind of polynomial which when factored, will always result to a binomial squared. Naturally then, if you square a binomial, the answer you get will always be a perfect square polynomial. The formula to square a binomial is as follows:

(x + y)^2 = x^2 + 2xy + y^2

You may notice that the first and last values of the answer are just the two values within the bracket squared respectively. The center value is just the two values in the brackets multiplied together and then multiplied by 2. If you FOILed the question instead:

(x + y)^2 = (x + y)(x + y)

= x^2 + xy + xy + y^2

= x^2 + 2xy + y^2

Example:

(2x + 3y)^2

Squaring the first and last values:

= (2x)^2 + ? + (3y)^2

= 4x^2 + ? + 9y^2

Multiplying the two values together and by 2:

= 4x^2 + 2(2x)(3y) + 9y^2

= 4x^2 + 12xy + 9y^2

When factoring, you can tell if a polynomial is a perfect square by using these same parameters. The first and last values will be perfect squares and the center value will be the square roots of those values multiplied by each other and 2. If you know that the polynomial is a perfect square, you can just write the roots of the first and last values in a set of brackets and square it all.

Example:

25x^2 + 10xy + y^2

25x^2 and y^2 are both perfect squares, their square roots are 5x and y respectively.

2(5x)(y) = 10x, so this is a perfect square polynomial.

Therefore,

25x^2 + 10xy + y^2

= (5x + y)^2