Let’s suppose you’ve got a polynomial function on your hands, something like this:

f(x) = 3x^4 – 11x^3 +10x – 4

And you have to find the zeros of the function, AKA where it intersects the x axis. Well, this ain’t a quadratic so using the quadratic formula is out. You could try to factor it… but ain’t nobody got time for that. We could just start plugging in random numbers for x and see if we ever get zero, but there sure are a lot of numbers. Same thing with trying long or synthetic division. May I present to you…

It just says that if your polynomial has all integer coefficients (that means the numbers in front of the x’s are all whole numbers), then if this polynomial has any rational (that means fraction) roots then those roots will have a factor of the constant term in the numerator, and a factor of the leading coefficient in the denominator.

uhhhhh. wut?

Let’s look at f(x) = 3x^4 – 11x^3 +10x – 4 again. All the coefficients are integers, so we can apply RRT. The constant term is -4. So any rational root of f(x) will have 1, 2 or 4 in the numerator. Leading coefficient is 3, so we’re looking at a denominator of 1 or 3 for a possible root. Now we can make a list of possible rational roots:

1, 2, 4, 1/3, 2/3, 4/3, -1, -2, -4, -1/3, -2/3, -4/3

Now we’ve narrowed down the possibilities A LOT. Use these possible roots with long/synthetic division, or just plug them into the function and see if you get zero, to find out which ones (if any) are roots of f(x).

Any questions?

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The English language is full of words that sound similar, but are spelled differently and mean completely different things. However, none are more infamous than the trio of “there,” “their,” and “they’re.” It’s a common source of frustration for many English speakers, but don’t worry, we’ll save you a few headaches with these tips.

Whenever you’re talking about physical space, you need to use “there.” It’s what allows use to visualize where things are located. For example, “Put the boxes over there.” You know “there” is referring to a place.

One way to think about “their” is as the plural form of “your.” It shows possession of something, but instead of to one person, to a group of people or even an organization. For example, “That painting belongs to their collection.”

This one is simple. “They’re” is an adjective that describes an organization or group of people and literally translates to “they are.” Here’s an example of how it works: “They’re the most successful coffee shop in town.”

So how do you tell the difference between all three. Well, it’s always easy to know when to use “they’re” because all you have to do is substitute it with “they are.” If the sentence still makes sense then you’re using it properly!

As for the other two, you simply have to make an effort to memorizing which one refers to what. **One neat trick for “there” is to associate it with “where.”** The two are spelled similarly and when someone asks “where?” your response is “there.” Pretty easy, right?

If you can remember these two rules then, by process of elimination, you’ll know when you’re supposed to use “their.” Once you get the hang of it, you’ll use these tests less and less until you eventually learn to recognize each by their spelling.

]]>Let’s say you’ve got some crazy trig equation that you have to solve… something like this…

tan^2(x) + tan(x) = 6

And you’ve got to figure out what x is. So you use all kinds of different trig identities, going around and around in circles trying to make the expression look nice. But that is getting you nowhere. How about try this. Let tan(x)=t. Then substitute. What you get is this:

tan^2(x) + tan(x) = 6

t^2+t=6

Subtract 6 from both sides, and then you’ve got a quadratic that you can solve by factoring or using the quadratic formula. You know how to solve quadratics, right? Right.

t^2+t=6

t^2+t-6=0

(t-2)(t+3)=0

t-2=0 or t+3=0

t=2 or t=-3

Now substitute tangent back in for t, take the inverse tangent of both sides, and you get you answer! This won’t work for every trig equation you have to solve, but its a handy dandy little trick you can keep in your back pocket for when you see a trig function squared.

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It’s not always your fault that you’re not aware of some of these grammar rules. (See what I did there? Hint: your vs. you’re) English is a difficult language to master, especially if it’s not your native tongue. With a few simple tricks, however, you’ll never confuse “your” and “you’re” again.

It’s easy to know when you’re using “you’re” correctly. All you have to do is ask yourself, can I substitute “you’re” with “you are” in this sentence? If the answer is yes, then you’re using “you’re” the right way. If the answer is no, then you want to use “your” instead.

It’s pronounced the same way as “you’re,” but “your” is only used to show that something belongs to someone. For example, “I’ll take your dog for a walk.” The “your” shows that the dog belongs to the second person in the sentence. This is the proper way to use “your.”

**You could also do the “you are” test**. If you substitute “you are” for “your” then the sentence doesn’t make sense: “I’ll take you are dog for a walk” (incorrect). Since the “you are” test failed, then you know you must use “your” instead of “you’re.”

This may seem like a minor mistake, but seeing an error like this on an essay or, worse yet, your personal statement is a huge red flag. Use the techniques shown above to always sound intelligent in your writing. Also, feel free to checkout Brightstorm Grammar for more tips.

]]>Successful college applications have identities. They have common themes that are visible throughout every part of the application. It’s not enough to simply answer all the questions, you have to express your interests and personality wherever possible. Here are three ways to give your applications a clear identity.

If you want to be an English major, for example, not only do you have to avoid spelling errors on your application like everyone else, but you also need to reflect strong writing skills in all your answers. Furthermore, you need to write a powerful essay that keeps readers engaged.

If you want to major in engineering or computer science, make sure you’ve taken all the necessary math and science courses. Also, your essay should reinforce your passion for the field and what you hope to achieve with your degree.

Lots of students like to mention that they’re hard workers who are dedicated to learning. However, it doesn’t mean much unless you have the experience to show for it. If your angle is being an overachiever, make sure you have lots of leadership experience and extracurricular activities to show for it.

These are just examples for how to structure your college application. Every application should be unique and play to your strengths. But most importantly, create a theme that runs through the entire application, including the essay!

Our final post in this series covers the last key ingredients you need to make your college application stand out. You can also check out Brightstorm College Advice for more topics by our admissions expert.

]]>The last thing you want on your college application is a spelling error, or even worse, an unanswered question! The best way to avoid big mistakes is to print and review your application before submitting it. Here are three steps you should take.

Before you submit your college application, there’s usually an option to print it out first. All applications on the web have this feature, some just have it in different places than others. Find this option and print out the entire application, but be careful not to submit it just yet.

The best way to check for errors in your college application is to read the whole thing out loud. Relax, it’s not as bad as you think. It might have taken you a while to fill out your application, but it will only take you a fraction of the time to read it all. Plus, you wouldn’t want silly spelling errors on it, would you?

The final measure in making sure there are absolutely no mistakes on your college application is to ask one of your parents or friends to review it for you. Since they don’t have the same kind of investment or attachment to it, it’ll be easier for them to point out parts that don’t make sense.

Follow all the strategies in this series and you’ll have a much better chance of making your college application stand out. If you haven’t already done so, go back and read the other posts in this series. Also, be sure to check out Brightstorm College Advice for more topics by our admissions expert.

]]>When it comes to writing well, it’s the small things that matter most. You may not think that a simple mistake, such as confusing “its” and “it’s” would be a big deal, but it is. It can mean the difference between getting into your college of choice and receiving the job offer of your dreams.

In this series, we’re going to explore all the common mistakes people make when writing and how to correct them. Since I already mentioned it, let’s start with the difference between “its” and “it’s.”

The reason most people confuse “its” and “it’s” is that, in English, the apostrophe is usually used to show possession. For example, when you want to say that a car belongs to Mike, you say “that’s Mike’s car.” The apostrophe shows that the car is the property of Mike.

However, don’t forget that apostrophes are also used for contractions. For example, instead of saying “do not,” you can simply use an apostrophe and say “don’t.” The two mean the same thing. It may be a little confusing, but this is proper way to use “it’s.”

Whenever you use “it’s” make sure you mean to say “it is.” For example, instead of saying “it is cold outside” it’s more natural to simply say “it’s cold outside.” This is the proper way to use “it’s.”

It may look weird since you’re used to showing possession with an apostrophe, such as “Mike’s car,” but when you want to show possession using “it” you MUST use “its.” For example, “it’s Mike’s car, its color is red.” Do you see how “it’s” and “its” is used in this sentence? You can always expand “it’s” to “it is,” but “its” always remains the same.

**Good:** It is Mike’s car, its color is red.

**Bad:** It is Mike’s car, it is color is red.

Whenever you write, always ask yourself: Can I substitute it for “it is.” If the answer is yes the you want to use “it’s.” If the answer is no, then you want to use “its.”

Get more helpful writing tips with our next post which focuses on the difference between “your” and “you’re.”

]]>Submitting your college application early will give you a big advantage over other applicants. Especially if the college does rolling admissions. But if you’re going to submit early, here are a few things you should be aware of.

The quality of your application is infinitely more important the the submission date. Sending off a sloppy application early won’t win you any points. You’ll have a better chance of getting accepted with a well-organized application than being the first to submit.

If the college does rolling admissions, it means there will be more spaces available the earlier you submit. As availability becomes more scarce, colleges with rolling admissions are more likely to scrutinize your application and reserve spaces for exceptional students only.

Even if the college doesn’t do rolling admissions, submitting your application early shows that you’re a responsible student. It will definitely make your college application stand out and give you more time to focus on other things afterwards.

The best part about submitting your application early is not having to worry about the system crashing on the last day. Often times, college web servers get so overloaded by students trying to submit their college applications on the last day that they crash and prevent you from submitting on time. Submit early to avoid this disaster!

Submitting your college application early is one of the most important aspects of a successful application. Read our next post for the third key ingredient to making your college application stand out. Don’t forget to also check out Brightstorm College for more topics by our admissions expert.

]]>The biggest and easiest mistake to make on your college application is to not follow directions. In this series, we’re going to provide you with the most awesome and necessary strategies for making sure your college application gets noticed by your top university choices. The first strategy is to simply follow directions.

Failing to follow directions is the fastest way to get your application tossed into the rejection pile. Why? Because it shows carelessness and a lack of attention to detail. No one enjoys filling out applications, but that doesn’t give you a free pass to rush through them.

Read through the instructions for each section carefully so that you’re filling them out properly. This includes answering all the questions that apply to you and making sure your response accurately addresses the question.

In addition to following the instructions spelled out in each section, it’s also important to follow all English grammar rules. This includes using proper capitalization and writing complete sentences. This may seem like a no-brainer, but you’d be surprised how many people forget to do this.

Following directions is only the first step to grabbing the admissions officer’s attention. Read our next post for a strategy that can give you a huge advantage over other applicants. Also, check out Brightstorm College for other great topics by our admissions expert.

]]>The last thing you need to know to make algebra II your best friend is how imaginary numbers work. They might sound really abstract at first, but they’re not. In fact, imaginary numbers, which are represented as *i*, have a really specific definition.

So what’s the big deal with imaginary numbers? Well, they’re really important for finding the roots of polynomials and solving complex trigonometry and calculus problems. But for now, all you need to know is that *i *works just like *x*.

The distributive law, FOIL, and all other math laws work exactly the same for *i* as they do for *x*, except whenever we see *i²* we substitute it with -1. That’s it!

Now that you know how functions, graphing inequalities, and imaginary numbers work, you’ll have no problem becoming best friends with algebra II. For more explanations about these topics, read the rest of this series and check out Brightstorm Algebra 2.

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