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Difference of Perfect Squares - Concept
Simplifying rational expressions combines everything learned about factoring common factors and polynomials. When simplifying rational functions, factor the numerator and denominator into terms multiplying each other and look for equivalents of one (something divided by itself). Include parenthesis around any expression with a "+" or "-" and if all terms cancel in the numerator, there is still a one there.
When you're asked to factor sometimes it can be a real drag so you want to look for any special shortcuts you can.
One type of shortcut involves what we call "difference of perfect squares," so in order to identify whether or not you can use this shortcut, first you have to know what a difference of perfect squares mean. First thing, what does difference mean? Do you remember? Difference means you're subtracting two terms so let's check it out. Once you have a subtraction going on, a difference of perfect squares means that first of all the coefficients are perfect squares, the constants are perfect squares and also the powers on the variable terms are even.
Let me show you some examples of things that are perfect squares. Something like this, 16x squared take away 81, I have a coefficient that's a perfect square, I have a constant that's a perfect square and I have an exponent that's even a power that's even. Something else, I could have x squared take away 4y squared, I have exponents that are even it's a difference problem I have my coefficients that are perfect squares that's a difference of perfect square. Here's one more for you, 4y squared take away 9x to the sixth, that's kind of weird because we have x to the sixth but it's okay as long as the powers are even. I have two different variables they both have even exponents and then my coefficients are perfect squares. So these do represent the difference of perfect squares and a factoring shortcut.
I'm going to show you some that are not difference of perfect squares. Something like this, If I took 16x squared plus 81, it's almost the same as this only look it's not a difference anymore this is a sum so this is not a difference of perfect squares, something else I could have would be 16x squared take away 8y squared, this is not a difference of perfect squares because my coefficient right there is not a perfect square, 8 is not a perfect square that's why this guy is not a difference of perfect squares. Here's one more that's not a difference of perfect squares cause I have that 8 again, 8 as a coefficient or as a constant is not a perfect square so it doesn't count. These guys are differences of perfect squares and here is why we talk about that like here's that's all the build up, here's like the big sh-bang when you have a difference of perfect square squares and you're asked to factor, your factor form looks like this, a-b times a+b is equal to the difference of perfect squares a squared take away b squared, so in your situation you already know how to multiply stuff so you could do this multiplying product to get this answer but in your homework problems you're going to be doing it backwards they're going to give you something like this or like this like this like this, they're going to be asking you to write it in the factored form, so we're going to go through and look at a couple of examples just keep in mind this shortcut only works if you meet these three criteria you have a difference problem and you have perfect square coefficients and constants and the powers on both variables have to be even.