Polynomials can also be used to do approximation, something that's usually covered in Calculus classes. Shudder. If you go here, you'll see polynomials whose shape become closer to what we want as we let the degree increase. These approximating polynomials are called Taylor polynomials, and they're used for everything from making your calculator work to storing music on CDs. You could even store a Taylor Swift CD using Taylor polynomials. We'd keep going, but it's probably a better idea not to.
The integers form something called a group, and single-variable polynomials with integer coefficients also form a ring. Fortunately, there's no creepy little girl climbing out of your television set for this one.
The rational numbers form a field, as do the rational expressions. A field is a collection of things in which the things "behave very nicely." In a field we can add, subtract, multiply, and divide. There's something called 0 (you may have heard of it) that we can add to any element of the field without changing the value of that element, and there's something called 1, which may also ring a bell, that we can multiply by any element of the field without changing the value of the element. There are also a few other things that must be true in a field. You'll have a field day with this one.
While there are infinitely many rational numbers and infinitely many rational expressions, it's also possible to have a field with only a finite number of things. This is always nice, as it gives us something to count when we're bored without going out of our minds. Using clock arithmetic (also known as modular arithmetic) on a clock that has a prime number of numbers (for example, a clock with the seven numbers 0, 1, 2, 3, 4, 5, 6) we have a finite collection of numbers that acts like the infinite collection of rational numbers. Good luck trying to meet someone on time when they ask you to be somewhere at 7, though.
There are three steps to solving a math problem.
A man gave his three sons a herd of camels. He said that half of the camels should go to the eldest son, one-third of the camels should go to the middle son, and one-ninth of the camels should go to the youngest son.
Ugh, he probably divided them unfairly to start a fight. What a dromedary queen.
After dividing the camels, the sons found that there were two camels left over. How many camels were in the original herd? Also, is it true that the oldest brother titled his autobiography My Humps?
1. Figure out what the problem is asking.
This part is fairly straightforward. Some camels got divided into groups, and we want to know how many total camels there are.
2. Solve the problem.
We'll translate the problem slowly from English into math. We know that:
Since we know how many camels each son receives in terms of the total number of camels, it makes sense to have a variable for the total number of camels. We'll use c, for camel. Reading the problem again, we can translate each piece of the equation into symbols:
The overall equation translates into symbols as:
From here, we know what to do. Isn't that the best feeling? We solve the equation using either of the two methods we learned earlier. Right now we'll do it by eliminating denominators, because that's what our Magic 8-Ball advised us to do.
First multiply both sides of our equation by 18 (the LCD of 2, 3, and 9) to find:
18c = 9c + 6c + 2c + 36
Simplify to get:
c = 36
And there we are.
3. Check the answer.
If there were originally 36 camels, then we should be able to give half of 36 to the eldest son, one-third of 36 to the middle son, one-ninth of 36 to the youngest son, and have 2 left over.
We do have 2 left over after the sons receive their camels, since 36 – 34 = 2.
On an entirely unrelated subject, if you're looking for camels to foster, please contact us here at Shmoop. We're determined to find them a good home. Warning: they spit.