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A Comprehensive Factoring Practice Quiz.

This is a comprehensive factoring practice page. Each of the problems below are randomly generated and randomly drawn from all the different types of factoring we have learned in the polynomial chapter. Each time you hit the “Another” button in the top right corner, the type of technique necessary and the values of each of the problems will change randomly.

I highly encourage you to keep redoing this page by hitting that “Another” button in the top right until you are able to answer each of the problems relatively easily and can quickly recognize which technique to use on any given problem.

Note: There is currently a known bug that Xronos may not correctly mark a factorization if it has a negative sign on the outside of the parentheses. To avoid this, simply multiply the negative value into one of your parentheses.

Hint: Rather than give individual hints (which can be challenging given the dynamically generated nature of the content below) I will give an overall hint here; note that this applies to all of the below. Remember that your aim is to fully factor the polynomial, but the techniques you may need to can be different from problem to problem. For that reason it is helpful to think of a general flow of techniques, starting from easiest (to do, or to rule out as possible) to hardest/longest to complete. I would recommend the following order, but this is entirely personal preference, so use the order that works best for you:

(a)
The first step of any factoring process should always be to factor out any common terms! Even if it is just a constant, this will make the constants/coefficients you have to deal with later much easier to deal with as they will be smaller. Trust me, it makes a big difference!
(b)
Quadratic Forms: It is usually easy to tell if a polynomial is a quadratic form as it must be 2 or 3 terms, and have that very specific format of $ax^{2n} + bx^n + c$ (i.e. that the leading term’s power is exactly twice the only other power of $x$, and the last term is a constant, possibly $0$). For this reason, you can usually see if any of the following techniques are even possible with the polynomial you are trying to factor:
• Factoring Coefficients
• AC-Method
• Difference of Squares (or cubes, although that isn’t a quadratic form technique, technically).
• Completing the Square
(c)
Next, if you can’t use any of the quadratic form techniques, a good option is to see if you can factor by grouping. Remember this requires a non-prime number of terms generally (i.e. you need something like 4, 6, or 8 terms) to have any hope of doing a grouping method.
(d)
If you can’t use any of the above methods, Rational Root Theorem is your tool of last resort. Remember Rational Root Theorem is terrible! It takes a while to do, and it’s basically a better version of guess and check; which is always a method of last resort in math. Nonetheless, it is sometimes all you can do.

Finally, once you have found a factor that you can pull out of the polynomial, treat the resulting “chunk” of a polynomial that you have left (after pulling out the factor you found) as a new problem; start your process over again by trying to factor out any common terms, looking at quadratic form techniques, factor by grouping, etc. The goal is to use the easier techniques if at all possible, at any stage; using rational root theorem a bunch of times in a row is going to take way too long on an exam, and realistically you rarely need to use it more than a couple times (at most) for a given problem.

1 : Factor the following polynomial completely.

2 : Factor the following polynomial completely.

3 : Factor the following polynomial completely.

4 : Factor the following polynomial completely.

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6 : Factor the following polynomial completely.

7 : Factor the following polynomial completely.

8 : Factor the following polynomial completely.

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10 : Factor the following polynomial completely.

11 : Factor the following polynomial completely.

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15 : Factor the following polynomial completely.