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One of the common questions I get asked when I am tutoring algebra is how to find the difference of squares. First, what exactly is a difference of squares, and what is it used for? Second, how do you find it? The difference of squares is a tool used to factor certain types of polynomials. Factoring is often useful in simplifying equations and allow some form of cancellation or combination of like factors. The difference of squares will allow you to factor some polynomial types that are not otherwise factorable, making it a useful tool in algebra and anything that uses algebra. So how do you find it? You can find the difference of squares for any polynomials which is a difference of two perfect squares. Take the simplest case: x2 -1. This polynomial is the difference of two perfect squares: x2 is, obviously, the square of x while 1 is the square of 1. The resulting factors, using the difference of squares, is (x+1)(x-1).   To confirm that this is,... read more

I was working with a student today, and as we worked through the section in his book dealing with Trigonometric Identities and Pythagorean Identities, we stumbled across a problem that gave us a bit of trouble. The solution is not so complicated, but it sure had us stumped earlier.   The problem was presented as such:       Factor and simplify the following using Trigonometric and Pythagorean Identities:   sec3(x) - sec2(x) - sec(x) + 1       We tried a couple of different approaches, such as factoring sec(x) from each term:   sec(x) * [ sec2(x) - sec(x) - 1 + 1/sec(x) ]   and factoring sec2(x) from each term:   sec2(x) * [ sec(x) - 1 - 1/sec(x) + 1/sec2(x) ]   We followed these approaches through a few steps, but nothing we were attempting led to the solution. After doing some reading online, I found that the solution required a simple... read more

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