Here’s something nobody else in your Algebra II class (maybe even the teacher) will know. (Do not try to be the wise-cracker until you're ready to PROVE that this works!!)
Since first learning to factor, we've been told that "the rules” are... you
- CAN factor the DIFFERENCE OF TWO SQUARES (referred to by some texts as ‘DOTS’) into conjugates.....
x^2 - 16 = (x - 4)(x + 4)
- CAN NOT factor the SUM OF TWO SQUARES. This second admonition is true when the instructions say "factor completely across the integers" because, as you are about to see, the answers are NOT integers.
BUT You CAN factor the sum of two squares if you've studied imaginary numbers and know a simple little TRICK.
You will still use conjugates, but there needs to be an i in the second term of each binomial.....
x^2 + 16 = (x - 4i)(x + 4i)
Why does it work? Because when these two binomials are FOILed, the middle terms (O and I) cancel each other out. AND since i^2 is simply -1, the multiplication of the last terms gives you the task of subtracting a negative number, which is simplified to ADDING!!
Try factoring these sums of perfect squares and FOIL the binomials to verify that you actually CAN “break the old rules” when you know enough math!!
PROBLEM A: p2 + 64
PROBLEM B: 121 + y2 (hint: use the commutative property first and attach the i to the square root of 121)
PROBLEM C: x2 + 36 y2
And this concludes the demonstration of how a math nerd can break all the "rules" and get away with it, mathematically speaking!!