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A detailed tutorial on two-way counting. Step by step tutorial including several examples of two-way counting for reference.

Two-way counting is when any expression for a given quantity are determined using two different counting approaches. Many people believe that a quadratic equation is the perfect example of two-way counting, because you find the quantity in more than one way. However, this is incorrect. Two-way counting is actually a backwards method – you have the quantity already, you just need to figure out how you could get it. This is used often in combinations and permutations, where you often already know what quantity you need to have, you just have to figure out how to get there.

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A detailed tutorial on avoiding the freshman dream. Step by step tutorial including several examples of the freshman dream for reference.

The freshman dream is a mistake commonly made in algebra that was named for the probability that only freshman would make this mistake. In reality, this mistake can be made by anyone, regardless of your academic standing. The freshman dream is employed when you are given a squared binomial. If your equation looks like (x + n)^2, people using the freshman dream will write this as x^2 + n^2. However, this is wrong! Your equation should look like (x + n)(x + n) in the first step, and from there it is obvious to see that you would need to use FOIL to solve for it.

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A detailed tutorial on how to identify prime polynomials. Step by step tutorial including several examples of identifying prime polynomials for reference.

Prime polynomials are any polynomial that cannot be factored. Just like a number is prime if you can not break it down into two seperate whole numbers to multiply, a polynomial is prime if you cannot break it down into two separate binomials with whole numbers to multiply. When you run into a prime polynomial when trying to solve a quadratic equation, you cannot use the factoring method. what the factoring method does is split the polynomials into a binomial, which cannot be done to a prime polynomial. If you have a prime polynomial, you have to use the quadratic formula to solve it. At first, you can spot prime polynomials by attempting to factor it, but eventually you will be able to do it just by looking at it.

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A detailed tutorial on the solving of problems using the binomial theorem. Stepby step tutorial including several examples of how to solve problems using the binomial theorem for reference.

The binomial theorem is something you should all be familiar with – it is the alternative to the F.O.I.L. technique. It is used when you are given a binomial that is raised to a power. The simplest version of it is expressed like this:

This can also be expressed as a factorial notation, in the form:

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A detailed tutorial on the solving of conjugates and conjugate expressions. Step by step tutorial including several examples of how to solve conjugates and conjugate expressions for reference.

Conjugates are probably very familiar to you – if you have spent any time studying binomials, then you know what a conjugate is. However, there is one difference. A conjugate uses radicals, or square roots, instead of whole numbers. One number will be a whole number, and one number will be a radical for each binomial. You can solve them using the normal FOIL method that is used on binomials, and with the algebra tricks you learned for multiplying square roots.